Inhomogeneous minimization problems for the p(x) -Laplacian
aa r X i v : . [ m a t h . A P ] J a n INHOMOGENEOUS MINIMIZATION PROBLEMSFOR THE p ( x ) -LAPLACIAN CLAUDIA LEDERMAN AND NOEMI WOLANSKI
Abstract.
This paper is devoted to the study of inhomogeneous minimization problems associatedto the p ( x )-Laplacian. We make a thorough analysis of the essential properties of their minimizersand we establish a relationship with a suitable free boundary problem.On the one hand, we study the problem of minimizing the functional J ( v ) = R Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + fv (cid:17) dx . We show that nonnegative local minimizers u are solutions to the freeboundary problem: u ≥ P ( f, p, λ ∗ )) ( ∆ p ( x ) u := div( |∇ u ( x ) | p ( x ) − ∇ u ) = f in { u > } u = 0 , |∇ u | = λ ∗ ( x ) on ∂ { u > } with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) and that the free boundary is a C ,α surface with the exceptionof a subset of H N − -measure zero.On the other hand, we study the problem of minimizing the functional J ε ( v ) = Z Ω (cid:16) |∇ v | p ε ( x ) p ε ( x ) + B ε ( v ) + f ε v (cid:17) dx , where B ε ( s ) = R s β ε ( τ ) dτ , ε > β ε ( s ) = ε β ( sε ), with β a Lipschitz functionsatisfying β > , β ≡ , u ε are nonnegative local minimizers,then u ε are solutions to( P ε ( f ε , p ε )) ∆ p ε ( x ) u ε = β ε ( u ε ) + f ε , u ε ≥ . Moreover, if the functions u ε , f ε and p ε are uniformly bounded, we show that limit functions u ( ε →
0) are solutions to the free boundary problem P ( f, p, λ ∗ ) with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − M (cid:17) /p ( x ) , M = R β ( s ) ds , p = lim p ε , f = lim f ε , and that the free boundary is a C ,α surface with theexception of a subset of H N − -measure zero.In order to obtain our results we need to overcome deep technical difficulties and develop newstrategies, not present in the previous literature for this type of problems. Introduction
This paper is devoted to the study of inhomogeneous minimization problems associated to the p ( x )-Laplacian. We make a thorough analysis of the essential properties of their minimizers andwe establish a relationship with a suitable free boundary problem.The first minimization problem under consideration corresponds to the functional(1.1) J ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx. Key words and phrases.
Minimization problem, free boundary problem, variable exponent spaces, regularity ofthe free boundary, inhomogeneous problem, singular perturbation.2010
Mathematics Subject Classification.
In the particular case in which p ( x ) ≡ f ( x ) ≡
0, the functional becomes Z Ω (cid:16) |∇ v | λ ( x ) χ { v> } (cid:17) dx. The corresponding minimization problem in H (Ω) with prescribed nonnegative values on ∂ Ω wasfirst treated by Alt and Caffarelli in the seminal paper [2] motivated by the study of flow problemsof jets and cavities. In [2] it was shown that local minimizers are solutions of the following freeboundary problem: u ≥ ( ∆ u = 0 in { u > } u = 0 , |∇ u | = λ ∗ ( x ) on ∂ { u > } , with λ ∗ ( x ) = (2 λ ( x )) / and that the free boundary ∂ { u > } is a C ,α surface with the exceptionof a subset of H N − -measure zero.In the present work we prove that nonnegative local minimizers of functional (1.1) are solutionsto the inhomogeneous free boundary problem for the p ( x )-Laplacian: u ≥ P ( f, p, λ ∗ )) ( ∆ p ( x ) u := div( |∇ u ( x ) | p ( x ) − ∇ u ) = f in { u > } u = 0 , |∇ u | = λ ∗ ( x ) on ∂ { u > } , with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) .The p ( x )-Laplacian serves as a model for a stationary non-newtonian fluid with properties de-pending on the point in the region where it moves. For example, such a situation corresponds toan electrorheological fluid. These are fluids such that their properties depend on the magnitude ofthe electric field applied to it. In some cases, fluid and Maxwell’s equations become uncoupled anda single equation for the p ( x )-Laplacian appears (see [33]).The second minimization problem we deal with corresponds to the functional(1.2) J ε ( v ) = Z Ω (cid:16) |∇ v | p ε ( x ) p ε ( x ) + B ε ( v ) + f ε v (cid:17) dx, where B ε ( s ) = R s β ε ( τ ) dτ , ε > β ε ( s ) = ε β ( sε ), with β a Lipschitz function satisfying β > , β ≡ , u ε arenonnegative local minimizers to (1.2), then u ε are solutions to( P ε ( f ε , p ε )) ∆ p ε ( x ) u ε = β ε ( u ε ) + f ε , u ε ≥ u ε , f ε and p ε are uniformly bounded, we show that limit functions u ( ε →
0) are solutions to the free boundary problem P ( f, p, λ ∗ ) with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − M (cid:17) /p ( x ) , M = R β ( s ) ds , p = lim p ε , f = lim f ε .Problem P ε ( f ε , p ε ), when p ε ( x ) ≡ f ε ≡
0, arises in combustion theory to describe thepropagation of curved premixed equi-diffusional deflagration flames. The study of the limit ( ε → f ε p ε ( x )-Laplacian, this singular perturbation problem may model flame propagationin a fluid with electromagnetic sensitivity. NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 3 Our work here, for both minimization problems, consists in an exhaustive analysis of the prop-erties of nonnegative local minimizers, namely, global regularity and behavior close to the freeboundary. This analysis allows us to prove that nonnegative local minimizers u of (1.1), and func-tions u = lim u ε ( ε → u ε nonnegative local minimizers of (1.2), are weak solutions to thefree boundary problem P ( f, p, λ ∗ ) (Theorems 5.1 and 5.3).In order to obtain our results we need to overcome deep technical difficulties and develop newstrategies, not present in the previous literature for this type of problems.One of the results we would like to highlight is the proof of the Lipschitz continuity of nonnegativelocal minimizers of functional (1.1) (Theorem 3.3 and Corollary 3.2). Our proof relies on a carefulrescaling argument, which transforms the problem into a minimization problem for a more generaloperator with nonstandard growth for which the control of the coefficients becomes nontrivial. Thisresult, which is new for f
0, is also new in the homogeneous case f ≡ < p ( x ) < p ( x )-Laplacian are of particular interestin the range 1 < p ( x ) < P ε ( f ε , p ε ) and for weak solutions to the free boundary problem P ( f, p, λ ∗ ) we recentlyobtained in our works [24] and [25], respectively.As a consequence we derive the smoothness of the free boundary for nonnegative local minimizers u of (1.1). More precisely, we prove that the free boundary ∂ { u > } is a C ,α surface with theexception of a subset of H N − -measure zero (Theorem 5.2).In an analogous way, we get the smoothness of the free boundary for limit functions u ( ε → u ε of (1.2), i.e., the free boundary ∂ { u > } is a C ,α surface withthe exception of a subset of H N − -measure zero (Theorem 5.4).We also obtain further regularity results on the free boundary, for both minimization problems,under further regularity assumptions on the data (Corollaries 5.1 and 5.2). In particular, if thedata are analytic, the free boundary is an analytic surface with the exception of a subset of H N − -measure zero.As stated above, the minimization problem with the functional in (1.1) was first studied by Altand Caffarelli in [2] with p ( x ) ≡ f ≡
0. Still in the homogeneous case f ≡
0, the problemwas studied by Alt, Caffarelli and Friedman in [3] for a quasilinear equation in the uniformlyelliptic case, then the p -Laplacian ( p ( x ) ≡ p ) was treated in [11], an operator with power-likegrowth was studied in [27], and the case of a variable power p ( x ) was considered in [5]. The linearinhomogeneous case was treated in [16] and [19].We remark that the inhomogeneous minimization problem for functional (1.1) with f p ( x ) ≡ p = 2.On the other hand, as pointed out above, problem P ε ( f ε , p ε ) —arising in combustion theory—was first rigorously studied in [4] when p ε ( x ) ≡ f ε ≡
0. Since then, much research has beendone on this problem, see [6, 7, 9, 12, 20, 21, 28, 32, 34]. For the inhomogeneous case we refer to[22, 23, 29, 30]. Preliminary results for the p ε ( x )-Laplacian were obtained in [24]. CLAUDIA LEDERMAN AND NOEMI WOLANSKI
We also remark that the inhomogeneous minimization problem for functional (1.2) with f ε p ε ( x ) ≡ p ε = 2.When f ε ≡ p ε ( x ) p ε .An outline of the paper is as follows: In Section 2 we define the notion of weak solution to thefree boundary problem P ( f, p, λ ∗ ) and include some related definitions and results. In Section 3we prove existence of minimizers of the energy functional (1.1) and develop an exhaustive analysisof the essential properties of functions u which are nonnegative local minimizers of that energy. InSection 4 we prove existence of minimizers of the energy functional (1.2) and develop an analogousanalysis of the properties of functions u ε which are nonnegative local minimizers of that energy andmoreover, we get results for their limit functions u . Finally, in Section 5 we study the regularityof the free boundary for both minimization problems. We conclude the paper with an Appendixwhere we collect some results on variable exponent Sobolev spaces as well as some other resultsthat are used in the paper.1.1. Preliminaries on Lebesgue and Sobolev spaces with variable exponent.
Let p : Ω → [1 , ∞ ) be a measurable bounded function, called a variable exponent on Ω and denote p max =esssup p ( x ) and p min = essinf p ( x ). We define the variable exponent Lebesgue space L p ( · ) (Ω) toconsist of all measurable functions u : Ω → R for which the modular ̺ p ( · ) ( u ) = R Ω | u ( x ) | p ( x ) dx isfinite. We define the Luxemburg norm on this space by k u k L p ( · ) (Ω) = k u k p ( · ) = inf { λ > ̺ p ( · ) ( u/λ ) ≤ } . This norm makes L p ( · ) (Ω) a Banach space.There holds the following relation between ̺ p ( · ) ( u ) and k u k L p ( · ) :min n(cid:16) Z Ω | u | p ( x ) dx (cid:17) /p min , (cid:16) Z Ω | u | p ( x ) dx (cid:17) /p max o ≤ k u k L p ( · ) (Ω) ≤ max n(cid:16) Z Ω | u | p ( x ) dx (cid:17) /p min , (cid:16) Z Ω | u | p ( x ) dx (cid:17) /p max o . Moreover, the dual of L p ( · ) (Ω) is L p ′ ( · ) (Ω) with p ( x ) + p ′ ( x ) = 1.Let W ,p ( · ) (Ω) denote the space of measurable functions u such that u and the distributionalderivative ∇ u are in L p ( · ) (Ω). The norm k u k ,p ( · ) := k u k p ( · ) + k|∇ u |k p ( · ) makes W ,p ( · ) (Ω) a Banach space.The space W ,p ( · )0 (Ω) is defined as the closure of the C ∞ (Ω) in W ,p ( · ) (Ω).For the sake of completeness we include in an Appendix at the end of the paper some additionalresults on these spaces that are used throughout the paper.1.2. Preliminaries on solutions to p ( x ) -Laplacian. Let p ( x ) be as above, g ∈ L ∞ (Ω) and a ∈ L ∞ (Ω), a ( x ) ≥ a > u is a solution to(1.3) div( a ( x ) |∇ u ( x ) | p ( x ) − ∇ u ) = g ( x ) in Ωif u ∈ W ,p ( · ) (Ω) and, for every ϕ ∈ C ∞ (Ω), there holds that Z Ω a ( x ) |∇ u ( x ) | p ( x ) − ∇ u · ∇ ϕ dx = − Z Ω ϕ g ( x ) dx. NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 5 Under the assumptions of the present paper (see 1.3 below) it follows as in Remark 3.2 in [35] that u ∈ L ∞ loc (Ω).Moreover, for any x ∈ Ω, ξ, η ∈ R N fixed we have the following inequalities(1.4) | η − ξ | p ( x ) ≤ C ( | η | p ( x ) − η − | ξ | p ( x ) − ξ ) · ( η − ξ ) if p ( x ) ≥ , | η − ξ | (cid:16) | η | + | ξ | (cid:17) p ( x ) − ≤ C ( | η | p ( x ) − η − | ξ | p ( x ) − ξ ) · ( η − ξ ) if p ( x ) < , with C = C ( N, p min , p max ). These inequalities imply that the function A ( x, ξ ) = a ( x ) | ξ | p ( x ) − ξ isstrictly monotone. Then, the comparison principle for equation (1.3) holds on bounded domainssince it follows from the monotonicity of A ( x, ξ ).1.3. Assumptions.
Throughout the paper we let Ω ⊂ R N be a domain. Assumptions on p ε ( x ) and p ( x ) . We assume that the functions p ε ( x ) are measurable and verify1 < p min ≤ p ε ( x ) ≤ p max < ∞ , x ∈ Ω . For our main results we need to assume further that p ε ( x ) are uniformly Lipschitz continuous inΩ. In that case, we denote by L the Lipschitz constant of p ε ( x ), namely, k∇ p ε k L ∞ (Ω) ≤ L .Unless otherwise stated, the same assumptions above will be made on the function p ( x ).When we are restricted to a ball B r we use p − = p − ( B r ) and p + = p + ( B r ) to denote the infimumand the supremum of p ( x ) over B r .In some results we assume further that p ∈ W , ∞ (Ω) ∩ W ,q (Ω), for some q > Assumptions on λ ( x ) . We assume that the function λ ( x ) is measurable in Ω and verifies0 < λ min ≤ λ ( x ) ≤ λ max < ∞ , x ∈ Ω . In some results we assume that λ ( x ) is continuous in Ω and in our main results we assume furtherthat λ ( x ) is H¨older continuous in Ω. Assumptions on f ε ( x ) and f ( x ) . We assume that f ε , f ∈ L ∞ (Ω). In some results we assumefurther that f ∈ W ,q (Ω), for some q > Assumptions on β ε . We assume that the functions β ε are defined by scaling of a single function β : R → R satisfying:i) β is a Lipschitz continuous function,ii) β > ,
1) and β ≡ R β ( s ) ds = M .And then β ε ( s ) := ε β ( sε ) . CLAUDIA LEDERMAN AND NOEMI WOLANSKI
Notation. • N spatial dimension • Ω ∩ ∂ { u > } free boundary • | S | N -dimensional Lebesgue measure of the set S • H N − ( N − • B r ( x ) open ball of radius r and center x • B r open ball of radius r and center 0 • B + r = B r ∩ { x N > } , B − r = B r ∩ { x N < }• B ′ r ( x ) open ball of radius r and center x in R N − • B ′ r open ball of radius r and center 0 in R N − • – R – B r ( x ) u = | B r ( x ) | R B r ( x ) u dx • – R – ∂B r ( x ) u = H N − ( ∂B r ( x )) R ∂B r ( x ) u d H N − • χ S characteristic function of the set S • u + = max( u, u − = max( − u, • h ξ , η i and ξ · η both denote scalar product in R N • B ε ( s ) = R s β ε ( τ ) dτ Weak solutions to the free boundary problem P ( f, p, λ ∗ )In this section, for the sake of completeness, we define the notion of weak solution to the freeboundary problem P ( f, p, λ ∗ ) and we give other related definitions and results that we are goingto employ in the paper.We point out that in [25] we derived some properties of the weak solutions to problem P ( f, p, λ ∗ )and we developed a theory for the regularity of the free boundary for weak solutions.In this section p ( x ) will be a Lipschitz continuous function.We first need Definition 2.1.
Let u be a continuous and nonnegative function in a domain Ω ⊂ R N . We saythat ν is the exterior unit normal to the free boundary Ω ∩ ∂ { u > } at a point x ∈ Ω ∩ ∂ { u > } in the measure theoretic sense, if ν ∈ R N , | ν | = 1 andlim r → r N Z B r ( x ) | χ { u> } − χ { x / h x − x ,ν i < } | dx = 0 . Then we have
Definition 2.2.
Let Ω ⊂ R N be a domain. Let p be a measurable function in Ω with 1 < p min ≤ p ( x ) ≤ p max < ∞ , λ ∗ continuous in Ω with 0 < λ min ≤ λ ∗ ( x ) ≤ λ max < ∞ and f ∈ L ∞ (Ω). Wecall u a weak solution of P ( f, p, λ ∗ ) in Ω if(1) u is continuous and nonnegative in Ω, u ∈ W ,p ( · )loc (Ω) and ∆ p ( x ) u = f in Ω ∩ { u > } .(2) For D ⊂⊂ Ω there are constants c min = c min ( D ), C max = C max ( D ), r = r ( D ), 0 < c min ≤ C max , r >
0, such that for balls B r ( x ) ⊂ D with x ∈ ∂ { u > } and 0 < r ≤ r c min ≤ r sup B r ( x ) u ≤ C max . (3) For H N − a.e. x ∈ ∂ red { u > } (that is, for H N − -almost every point x ∈ Ω ∩ ∂ { u > } such that Ω ∩ ∂ { u > } has an exterior unit normal ν ( x ) in the measure theoretic sense) NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 7 u has the asymptotic development u ( x ) = λ ∗ ( x ) h x − x , ν ( x ) i − + o ( | x − x | ) . (4) For every x ∈ Ω ∩ ∂ { u > } ,lim sup x → x u ( x ) > |∇ u ( x ) | ≤ λ ∗ ( x ) . If there is a ball B ⊂ { u = 0 } touching Ω ∩ ∂ { u > } at x , thenlim sup x → x u ( x ) > u ( x )dist( x, B ) ≥ λ ∗ ( x ) . Definition 2.3.
Let v be a continuous nonnegative function in a domain Ω ⊂ R N . We say that v is nondegenerate at a point x ∈ Ω ∩ { v = 0 } if there exist c >
0, ¯ r > Z – B r ( x ) v dx ≥ cr for 0 < r ≤ ¯ r , (2.2) – Z – ∂B r ( x ) v dx ≥ cr for 0 < r ≤ ¯ r , (2.3) sup B r ( x ) v ≥ cr for 0 < r ≤ ¯ r . We say that v is uniformly nondegenerate on a set Γ ⊂ Ω ∩ { v = 0 } in the sense of (2.1) (resp.(2.2), (2.3)) if the constants c and ¯ r in (2.1) (resp. (2.2), (2.3)) can be taken independent of thepoint x ∈ Γ. Remark 2.1.
Assume that v ≥ ⊂ R N , v ∈ W ,p ( · ) (Ω) with ∆ p ( x ) v ≥ f χ { v> } , where f ∈ L ∞ (Ω), 1 < p min ≤ p ( x ) ≤ p max < ∞ and p ( x ) isLipschitz continuous. Then the three concepts of nondegeneracy in Definition 2.3 are equivalent(for the idea of the proof, see Remark 3.1 in [21], where the case p ( x ) ≡ f ≡ Energy minimizers of energy functional (1.1)In this section we prove existence of minimizers of the energy functional (1.1) and we develop anexhaustive analysis of the essential properties of functions u which are nonnegative local minimizersof that energy.We start with a definition and some related remarks Definition 3.1.
Let 1 < p min ≤ p ( x ) ≤ p max < ∞ , f ∈ L ∞ (Ω) and λ ( x ) measurable with0 < λ min ≤ λ ( x ) ≤ λ max < ∞ . We say that u ∈ W ,p ( · ) (Ω) is a local minimizer in Ω of J ( v ) = J Ω ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx if for every Ω ′ ⊂⊂ Ω and for every v ∈ W ,p ( · ) (Ω) such that v = u in Ω \ Ω ′ there holds that J ( v ) ≥ J ( u ). CLAUDIA LEDERMAN AND NOEMI WOLANSKI
Remark 3.1.
Let u be as in Definition 3.1. Let Ω ′ ⊂⊂ Ω and w − u ∈ W ,p ( · )0 (Ω ′ ). If we define¯ w = ( w in Ω ′ ,u in Ω \ Ω ′ , then ¯ w ∈ W ,p ( · ) (Ω) and therefore J ( ¯ w ) ≥ J ( u ). If we now let J Ω ′ ( v ) = Z Ω ′ (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx, it follows that J Ω ′ ( w ) ≥ J Ω ′ ( u ). Remark 3.2.
Let J be as in Definition 3.1. If u ∈ W ,p ( · ) (Ω) is a minimizer of J among thefunctions v ∈ u + W ,p ( · )0 (Ω), then u is a local minimizer of J in Ω.We first prove Theorem 3.1.
Assume that < p min ≤ p ( x ) ≤ p max < ∞ with k∇ p k L ∞ ≤ L , f ∈ L ∞ (Ω) and λ ( x ) is measurable with < λ min ≤ λ ( x ) ≤ λ max < ∞ . Let φ ∈ W ,p ( · ) (Ω) and assume that Ω is abounded domain. There exists u ∈ W ,p ( · ) (Ω) that minimizes the energy J ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx, among functions v ∈ W ,p ( · ) (Ω) such that v − φ ∈ W ,p ( · )0 (Ω) . Then, for every Ω ′ ⊂⊂ Ω there exists C = C (Ω ′ , k φ k ,p ( · ) , k f k L ∞ (Ω) , p min , p max , λ max , L ) such that (3.1) sup Ω ′ u ≤ C. Proof.
Let us prove first that a minimizer exists. In fact, let K = n v ∈ W ,p ( · ) (Ω) : v − φ ∈ W ,p ( · )0 (Ω) o . In order to prove that J is bounded from below in K , we observe that if v ∈ K , then J ( v ) ≥ p max Z Ω |∇ v | p ( x ) dx + Z Ω f v dx, and we have, by Theorem A.3 and Theorem A.4, Z Ω | f v | dx ≤ k f k p ′ ( · ) k v k p ( · ) ≤ k f k p ′ ( · ) ( k v − φ k p ( · ) + k φ k p ( · ) ) ≤ C k∇ v − ∇ φ k p ( · ) + C ≤ C k∇ v k p ( · ) + C . If (cid:16) R Ω |∇ v | p ( x ) dx (cid:17) /p min ≥ (cid:16) R Ω |∇ v | p ( x ) dx (cid:17) /p max we get, by Proposition A.1, Z Ω | f v | dx ≤ C (cid:16) Z Ω |∇ v | p ( x ) dx (cid:17) /p min + C ≤ C + 12 p max Z Ω |∇ v | p ( x ) dx. If, on the other hand, (cid:16) R Ω |∇ v | p ( x ) dx (cid:17) /p min < (cid:16) R Ω |∇ v | p ( x ) dx (cid:17) /p max , we get in an analogous way Z Ω | f v | dx ≤ C (cid:16) Z Ω |∇ v | p ( x ) dx (cid:17) /p max + C ≤ C + 12 p max Z Ω |∇ v | p ( x ) dx. NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 9 Taking C = max { C , C } , we get(3.2) J ( v ) ≥ − C + 12 p max Z Ω |∇ v | p ( x ) dx ≥ − C , which shows that J is bounded from below in K .At this point we want to remark that the constants C , ..., C above can be taken depending onlyon k φ k ,p ( · ) , k f k L ∞ (Ω) , p min , p max and L .We now take a minimizing sequence { u n } ⊂ K . Without loss of generality we can assume that J ( u n ) ≤ J ( φ ), so by (3.2), R Ω |∇ u n | p ( x ) ≤ C . By Proposition A.1, k∇ u n − ∇ φ k p ( · ) ≤ C and, as u n − φ ∈ W ,p ( · )0 (Ω), by Theorem A.4 we have k u n − φ k p ( · ) ≤ C . Therefore, by Theorem A.1 thereexist a subsequence (that we still call u n ) and a function u ∈ W ,p ( · ) (Ω) such that(3.3) || u || W ,p ( · ) (Ω) ≤ ¯ C, with ¯ C = ¯ C ( k φ k ,p ( · ) , k f k L ∞ (Ω) , p min , p max , λ max , L ) ,u n ⇀ u weakly in W ,p ( · ) (Ω) , and, by Theorem A.2, u n ⇀ u weakly in W ,p min (Ω) . Now, by the compactness of the immersion W ,p min (Ω) ֒ → L p min (Ω) we have that, for a subsequencethat we still denote by u n , u n → u in L p min (Ω) ,u n → u a.e. Ω . As K is convex and closed, it is weakly closed, so u ∈ K .It follows that λ ( x ) χ { u> } ≤ lim inf n →∞ λ ( x ) χ { u n > } , Z Ω λ ( x ) χ { u> } dx ≤ lim inf n →∞ Z Ω λ ( x ) χ { u n > } dx, lim n →∞ Z Ω f u n dx = Z Ω f u dx, Z Ω |∇ u | p ( x ) p ( x ) dx ≤ lim inf n →∞ Z Ω |∇ u n | p ( x ) p ( x ) dx. In order to prove the last inequality we observe that there holds(3.4) Z Ω |∇ u n | p ( x ) p ( x ) dx ≥ Z Ω |∇ u | p ( x ) p ( x ) dx + Z Ω |∇ u | p ( x ) − ∇ u · ( ∇ u n − ∇ u ) dx. Recall that ∇ u n converges weakly to ∇ u in L p ( · ) (Ω). Now, since |∇ u | p ( x ) − ∈ L p ′ ( · ) (Ω), byTheorem A.1 and passing to the limit in (3.4) we getlim inf n →∞ Z Ω |∇ u n | p ( x ) p ( x ) dx ≥ Z Ω |∇ u | p ( x ) p ( x ) dx. Hence J ( u ) ≤ lim inf n →∞ J ( u n ) = inf v ∈K J ( v ) . Therefore, u is a minimizer of J in K .Finally, in order to prove (3.1), we observe that, from Proposition A.1 and estimate (3.3), we havethat R Ω | u | p ( x ) dx ≤ ¯ C ( k φ k ,p ( · ) , k f k L ∞ (Ω) , p min , p max , λ max , L ). Thus, the desired estimate follows from the application of Proposition 2.1 in [35], since, by Lemma 3.1, ∆ p ( x ) u ≥ f ≥ −k f k L ∞ (Ω) inΩ. (cid:3) For local minimizers we first have
Lemma 3.1.
Let p, f and λ be as in Theorem 3.1. Let u ∈ W ,p ( · ) (Ω) be a local minimizer of J ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx. Then (3.5) ∆ p ( x ) u ≥ f in Ω . Proof.
In fact, let t > ≤ ξ ∈ C ∞ (Ω). Using the minimality of u we have0 ≤ t ( J ( u − tξ ) − J ( u )) ≤ t Z Ω (cid:16) |∇ u − t ∇ ξ | p ( x ) p ( x ) − |∇ u | p ( x ) p ( x ) (cid:17) dx − Z Ω f ξ dx ≤ − Z Ω |∇ u − t ∇ ξ | p ( x ) − ( ∇ u − t ∇ ξ ) · ∇ ξ dx − Z Ω f ξ dx and if we take t →
0, we obtain(3.6) 0 ≤ − Z Ω |∇ u | p ( x ) − ∇ u · ∇ ξ dx − Z Ω f ξ dx, which gives (3.5). (cid:3) Remark 3.3.
We are interested in studying the behavior of nonnegative local minimizers of theenergy functional (1.1).If u is as in Theorem 3.1 and we have, for instance, φ ≥ f ≤ u ≥ ξ = min( u, ∈ W ,p ( · )0 (Ω) so, for every0 < t < u − tξ ∈ φ + W ,p ( · )0 (Ω), with χ { u − tξ> } = χ { u> } . Then, in a similar way as in Lemma3.1, we get (3.6) and using that f ≤ R Ω |∇ ξ | p ( x ) dx = 0, which implies u ≥ u is any local minimizer of (1.1), the same argument employed in Theorem3.1 gives sup Ω ′ u ≤ C Ω ′ , for any Ω ′ ⊂⊂ Ω. Therefore, if u is any nonnegative local minimizer of(1.1), then u ∈ L ∞ loc (Ω).From now on we will deal with nonnegative local minimizers. Next we will prove that they arelocally Lipschitz continuous.First we need Lemma 3.2.
Let p and f be as in Theorem 3.1. Let Ω ⊂ (0 , d ) × R N − be a bounded domain.Assume a ∈ L ∞ (Ω) , a ( x ) ≥ a > , with k∇ a k L ∞ ≤ L . Let u ∈ W ,p ( · ) (Ω) be a solution to div (cid:0) a ( x ) |∇ u | p ( x ) − ∇ u (cid:1) = f in Ω with | u | ≤ M on ∂ Ω . Assume moreover that Ld < p min − .Then, there exists C = C ( M, p min , || f || L ∞ (Ω) , d, a , L, L ) such that | u | ≤ C in Ω .Proof. We consider, for α >
1, the function w ( x ) = M + e αd − e αx . Computing, we have w x i = − αe αx δ i , w x i x j = − α e αx δ i δ j , |∇ w | = αe αx . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 11 Therefore we obtaindiv (cid:0) a ( x ) |∇ w | p ( x ) − ∇ w (cid:1) = |∇ w | p ( x ) − h a ( x )∆ w + a ( x ) h∇ w, ∇ p i log |∇ w | + a ( x ) ( p ( x ) − |∇ w | X i,j w x i w x j w x i x j + h∇ w, ∇ a i i = a ( x )( αe αx ) p ( x ) − h − ( p ( x ) − α − p x ( x ) log( αe αx ) − a x ( x ) a ( x ) i ≤ a ( x )( αe αx ) p ( x ) − h − ( p min − α + L log α + Lαx + | a x ( x ) | a ( x ) i ≤ a ( x )( αe αx ) p ( x ) − h(cid:0) − ( p min −
1) + Ld (cid:1) α + L log α + L a i . If we let α ≥ α = α ( p min , d, a , L, L ) so that (cid:0) − ( p min −
1) + Ld (cid:1) α + L log α + L a <
0, we getdiv (cid:0) a ( x ) |∇ w | p ( x ) − ∇ w (cid:1) ≤ a α p min − h(cid:0) − ( p min −
1) + Ld (cid:1) α + L log α + L a i ≤ −|| f || L ∞ (Ω) , where the last inequality holds if we choose α ≥ α = α ( || f || L ∞ (Ω) , p min , d, a , L, L ).It follows that for α = max { α , α , } the corresponding function w satisfiesdiv (cid:0) a ( x ) |∇ w | p ( x ) − ∇ w (cid:1) ≤ −|| f || L ∞ (Ω) ≤ ± f in Ω . Since ± u ≤ w on ∂ Ω, we get ± u ≤ w ≤ M + e αd in Ω. This concludes the proof. (cid:3) Remark 3.4.
Let u be as in Lemma 3.2 in a domain Ω ⊂ ( − r, r ) × R N − . Then, defining¯ u ( x ) = u ( x − re ), ¯ a ( x ) = a ( x − re ), ¯ p ( x ) = p ( x − re ), ¯ f ( x ) = f ( x − re ) and ¯Ω = Ω + re , wehave div (cid:0) ¯ a ( x ) |∇ ¯ u | ¯ p ( x ) − ∇ ¯ u (cid:1) = ¯ f in ¯Ω. Then, the invariance by translations of the problem allowsus to apply Lemma 3.2 to ¯ u and conclude that, if L r < p min −
1, then | u | ≤ C in Ω, for a constant C = C ( M, p min , || f || L ∞ (Ω) , r, a , L, L ).Next, we prove that nonnegative local minimizers —of a more general functional than (1.1)—are locally H¨older continuous. Theorem 3.2.
Let p, f and λ be as in Theorem 3.1. Assume that < a ≤ a ( x ) ≤ a < ∞ , with k∇ a k L ∞ ≤ L . Let u ∈ W ,p ( · ) (Ω) ∩ L ∞ (Ω) be a nonnegative local minimizer of J a ( v ) = Z Ω (cid:16) a ( x ) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx and let B ˆ r ( x ) ⊂ Ω . Then, there exist < γ < and < ˆ ρ < ˆ r , ˆ ρ = ˆ ρ (ˆ r , N, p min , L ) and γ = γ ( N, p min ) , such that u ∈ C γ ( B ˆ ρ ( x )) . Moreover, k u k C γ ( B ˆ ρ ( x )) ≤ C with C depending onlyon N , ˆ r , p min , p max , L , λ max , k u k L ∞ ( B ˆ r ( x )) , k f k L ∞ ( B ˆ r ( x )) , a , a and L .Proof. We will prove that there exist 0 < γ < < ρ < r < ˆ r such that, if B r ( y ) ⊂ B ˆ r ( x )and ρ ≤ ρ , then(3.7) (cid:16) – Z – B ρ ( y ) |∇ u | p − dx (cid:17) /p − ≤ Cρ γ − , where p − = p − ( B r ( y )). Without loss of generality we will assume that y = 0. In fact, let 0 < r ≤ min { ˆ r , } , 0 < r ≤ r and v the solution of(3.8) div (cid:0) a ( x ) |∇ v | p ( x ) − ∇ v (cid:1) = f in B r , v − u ∈ W ,p ( · )0 ( B r ) . If r ≤ L ( p min − || v || L ∞ ( B r ) ≤ ¯ C with ¯ C = ¯ C ( L, p min , k u k L ∞ ( B ˆ r ( x )) , k f k L ∞ ( B ˆ r ( x )) , a , L ) . Let u s ( x ) = su ( x ) + (1 − s ) v ( x ). By using (3.8) and the inequalities in (1.4), we get(3.10) Z B r a ( x ) |∇ u | p ( x ) p ( x ) − a ( x ) |∇ v | p ( x ) p ( x ) + Z B r f ( u − v ) = Z dss Z B r a ( x ) (cid:16) |∇ u s | p ( x ) − ∇ u s − |∇ v | p ( x ) − ∇ v (cid:17) · ∇ ( u s − v ) ≥ C (cid:16) Z B r ∩{ p ≥ } a ( x ) |∇ u − ∇ v | p ( x ) + Z B r ∩{ p< } a ( x ) |∇ u − ∇ v | (cid:16) |∇ u | + |∇ v | (cid:17) p ( x ) − (cid:17) , where C = C ( p min , p max , N ).Therefore, by the minimality of u , we have (if A = B r ∩ { p ( x ) < } and A = B r ∩ { p ( x ) ≥ } ) Z A |∇ u − ∇ v | p ( x ) dx ≤ Cr N , (3.11) Z A |∇ u − ∇ v | ( |∇ u | + |∇ v | ) p ( x ) − dx ≤ Cr N , (3.12)where C = C ( p min , p max , N, λ max , a ).Let ε >
0. Take ρ = r ε and suppose that r ε ≤ /
2. Take 0 < η < A and (3.12), we obtain(3.13) Z A ∩ B ρ |∇ u − ∇ v | p ( x ) dx ≤ Cη /p min Z A ∩ B r ( |∇ u | + |∇ v | ) p ( x ) − |∇ u − ∇ v | dx + Cη Z B ρ ∩ A ( |∇ u | + |∇ v | ) p ( x ) dx ≤ Cη /p min r N + Cη Z B ρ ∩ A ( |∇ u | + |∇ v | ) p ( x ) dx. Therefore, by (3.11) and (3.13), we get(3.14) Z B ρ |∇ u − ∇ v | p ( x ) dx ≤ Cη /p min r N + Cη Z B ρ ∩ A ( |∇ u | + |∇ v | ) p ( x ) dx, where C = C ( p min , p max , N, λ max , a ).Since, |∇ u | q ≤ C ( |∇ u − ∇ v | q + |∇ v | ) q ), for any q >
1, with C = C ( q ), we have, by (3.14),choosing η small, that(3.15) Z B ρ |∇ u | p ( x ) dx ≤ Cr N + C Z B ρ |∇ v | p ( x ) dx, where C = C ( p min , p max , N, λ max , a ).Now let M ≥ || v || L ∞ ( B r ) ≤ M and define w ( x ) = v ( rx ) M in B . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 13 Then, there holds thatdiv (cid:0) ¯ a ( x ) |∇ w | ¯ p ( x ) − ∇ w (cid:1) = (cid:16) rM (cid:17) p ( rx ) − rf ( rx ) + r log (cid:16) rM (cid:17) a ( rx ) ∇ p ( rx ) · ∇ w ( x ) |∇ w ( x ) | p ( rx ) − in B , with ¯ p ( x ) = p ( rx ) and ¯ a ( x ) = a ( rx ). That is,div (cid:0) ¯ a ( x ) |∇ w | ¯ p ( x ) − ∇ w (cid:1) = B ( x, ∇ w ( x )) in B , with | B ( x, ∇ w ( x )) | ≤ C (cid:16) |∇ w ( x ) | ¯ p ( x ) (cid:17) in B , where C = C ( L, M, k f k L ∞ ( B ˆ r ( x )) , a ).From Theorem 1.1 in [14], it follows that w ∈ C ,α loc ( B ) for some 0 < α < B / |∇ w | ≤ C ( L, M, k f k L ∞ ( B ˆ r ( x )) , p min , p max , N, a , a , L ) , which implies(3.16) sup B r/ |∇ v | ≤ CMr .
Therefore, from (3.15) and (3.16), we deduce that(3.17) Z B ρ |∇ u | p ( x ) dx ≤ Cr N + Cρ N r − p + , with p + = p + ( B r ) and C = C ( L, k u k L ∞ ( B ˆ r ( x )) , k f k L ∞ ( B ˆ r ( x )) , λ max , p min , p max , N, a , a , L ).Here we have used the bound in (3.9).Then, if we take ε ≤ p min N , we have by (3.17) and by our election of ρ , that– Z – B ρ |∇ u | p − dx ≤ – Z – B ρ |∇ u | p ( x ) dx + 1 | B ρ | Z B ρ ∩{|∇ u | < } |∇ u | p − dx ≤ – Z – B ρ |∇ u | p ( x ) dx + 1 ≤ C (cid:16) rρ (cid:17) N + Cr − p + ≤ Cr − εN + Cr − p + ≤ Cr − p + = Cρ − p +(1+ ε ) . Now let r ≤ r ( ε, p min , L ) so that p + p − = p + ( B r ) p − ( B r ) ≤ ε , and small enough so that, in addition, r ε ≤ /
2. Then, if ρ ≤ ρ = r ε ,– Z – B ρ |∇ u | p − dx ≤ Cρ − (1+ ε ε ) p − = Cρ − (1 − γ ) p − , where γ = ε (1+ ε ) = γ ( N, p min ). That is, if ρ ≤ ρ = r ε (cid:16) – Z – B ρ |∇ u | p − dx (cid:17) /p − ≤ Cρ γ − . Thus (3.7) holds, with C = C ( L, k u k L ∞ ( B ˆ r ( x )) , k f k L ∞ ( B ˆ r ( x )) , λ max , p min , p max , N, a , a , L ).Applying Morrey’s Theorem, see e.g. [26], Theorem 1.53, we conclude that u ∈ C γ ( B ρ ( x )) and k u k C γ ( B ρ / ( x )) ≤ C for C = C (ˆ r , L, k u k L ∞ ( B ˆ r ( x )) , k f k L ∞ ( B ˆ r ( x )) , λ max , p min , p max , N, a , a , L ). (cid:3) As a corollary we obtain
Corollary 3.1.
Let u be as in Theorem 3.2. Then u ∈ C γ (Ω) for some < γ < , γ = γ ( N, p min ) .Moreover, if Ω ′ ⊂⊂ Ω , then k u k C γ (Ω ′ ) ≤ C with C depending only on N , dist(Ω ′ , ∂ Ω) , p min , p max , L , λ max , k u k L ∞ (Ω) , k f k L ∞ (Ω) , a , a and L . Then, under the assumptions of the previous corollary we have that u is continuous in Ω andtherefore, { u > } is open. We can now prove the following property for nonnegative local mini-mizers of (1.1) Lemma 3.3.
Let p, f and λ be as in Theorem 3.1. Let u ∈ W ,p ( · ) (Ω) ∩ L ∞ (Ω) be a nonnegativelocal minimizer of J ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx. Then (3.18) ∆ p ( x ) u = f in { u > } . Proof.
From Lemma 3.1 we already know that (3.5) holds. In order to obtain the opposite inequalityin { u > } , we let 0 ≤ ξ ∈ C ∞ ( { u > } ) and consider u − tξ , for t <
0, with | t | small.Using the minimality of u we have0 ≥ t ( J ( u − tξ ) − J ( u )) = 1 t Z Ω (cid:16) |∇ u − t ∇ ξ | p ( x ) p ( x ) − |∇ u | p ( x ) p ( x ) (cid:17) dx − Z Ω f ξ dx ≥ − Z Ω |∇ u − t ∇ ξ | p ( x ) − ( ∇ u − t ∇ ξ ) · ∇ ξ dx − Z Ω f ξ dx and if we take t →
0, we obtain0 ≥ − Z Ω |∇ u | p ( x ) − ∇ u · ∇ ξ dx − Z Ω f ξ dx, which gives the desired inequality, so (3.18) follows. (cid:3) We will make use of the following version of Harnack’s inequality
Proposition 3.1.
Let x ∈ R N and < δ ≤ . Let < p min ≤ p ( x ) ≤ p max < ∞ in B δ ( x ) ,with k∇ p k L ∞ ( B δ ( x )) ≤ L and f ∈ L ∞ ( B δ ( x )) . There exists a constant C > such that, if u ∈ W ,p ( · ) ( B δ ( x )) ∩ L ∞ ( B δ ( x )) is a nonnegative solution of ∆ p ( x ) u = f in B δ ( x ) , then, (3.19) sup B δ ( x ) u ≤ C (cid:2) inf B δ ( x ) u + δ (cid:3) . The constant C depends only on N , p min , p max , L , k f k L ∞ ( B δ ( x )) and k u k p δ + − p δ − L ∞ ( B δ ( x )) , where p δ + =sup B δ ( x ) p ( x ) and p δ − = inf B δ ( x ) p ( x ) . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 15 Proof.
We will first assume that x = 0 and δ = 1. From Theorem 1.1 in [14], we know that u ∈ C ( B (0)).Let y ∈ B / (0). Since ∆ p ( x ) u = f in B (0), by Theorem 2.1 in [35], applied in B / ( y ), we get(3.20) sup B ( y ) u ≤ C (cid:2) inf B ( y ) u + 1 (cid:3) , where C is a positive constant that can be chosen so that C > N , p min , p max , L , k f k L ∞ ( B (0)) and k u k p − p − L ∞ ( B (0)) , where p = sup B (0) p ( x ) and p − = inf B (0) p ( x ).We now cover B / (0) with k balls centered in B / (0) of radius 1 /
32 ( k ≥ x, y ∈ B / (0), we choose balls of the covering and points, and we number them, in such a waythat x = x ∈ B , x i ∈ B i ∩ B i +1 and x j = y ∈ B j , for 1 ≤ i ≤ j − j ≤ k .It follows from (3.20) that u ( x i ) ≤ C (cid:2) u ( x i +1 ) + 1 (cid:3) , i = 0 , . . . , j − , which gives u ( x ) ≤ C k (cid:2) u ( y ) + k (cid:3) . Therefore,∆ p ( x ) u = f in B (0) , implies(3.21) sup B (0) u ≤ C (cid:2) inf B (0) u + 1 (cid:3) , for a constant C > N , p min , p max , L , k f k L ∞ ( B (0)) and k u k p − p − L ∞ ( B (0)) .For general x ∈ R N and 0 < δ ≤
1, we take ¯ u ( x ) = δ u ( x + δx ). Then, as∆ ¯ p ( x ) ¯ u = ¯ f in B (0) , with ¯ p ( x ) = p ( x + δx ) and ¯ f ( x ) = δf ( x + δx ), there holds that ¯ u satisfies (3.21). Finally, observingthat p min ≤ ¯ p ( x ) ≤ p max in B (0), k∇ ¯ p k L ∞ ( B (0)) ≤ L , k ¯ f k L ∞ ( B (0)) ≤ k f k L ∞ ( B δ ( x )) , k ¯ u k ¯ p − ¯ p − L ∞ ( B (0)) = (cid:16) δ k u k L ∞ ( B δ ( x )) (cid:17) p δ + − p δ − , and (cid:16) δ (cid:17) p δ + − p δ − ≤ (cid:16) δ (cid:17) Lδ ≤ C ( L ) , we obtain the desired result. (cid:3) We will next prove the Lipschitz continuity of nonnegative local minimizers of (1.1). In the casein which f ≡ p ( x ) ≥ Theorem 3.3.
Let p, f, λ and u be as in Lemma 3.3. Let Ω ′ ⊂⊂ Ω . There exist constants C > , r > such that if x ∈ Ω ′ ∩ ∂ { u > } and r ≤ r then sup B r ( x ) u ≤ Cr.
The constants depend only on
N, p min , p max , L, || f || L ∞ (Ω) , λ min , λ max , || u || L ∞ (Ω) and dist(Ω ′ , ∂ Ω) . Proof.
Let us suppose by contradiction that there exist a sequence of nonnegative local minimizers u k corresponding to functionals J k given by functions p k , f k and λ k , with u k ∈ W ,p k ( · ) (Ω) ∩ L ∞ (Ω), p min ≤ p k ( x ) ≤ p max , k∇ p k k L ∞ ≤ L , || f k || L ∞ (Ω) ≤ M , λ min ≤ λ k ( x ) ≤ λ max , || u k || L ∞ (Ω) ≤ M andpoints ¯ x k ∈ Ω ′ ∩ ∂ { u k > } , such thatsup B rk/ (¯ x k ) u k ≥ kr k and r k ≤ k . Without loss of generality we will assume that ¯ x k = 0.Let us define in B , for k large, ¯ u k ( x ) = r k u k ( r k x ), ¯ p k ( x ) = p k ( r k x ), ¯ f k ( x ) = r k f k ( r k x ) and¯ λ k ( x ) = λ k ( r k x ). Then p min ≤ ¯ p k ( x ) ≤ p max , k∇ ¯ p k k L ∞ ( B ) ≤ Lr k , λ min ≤ ¯ λ k ( x ) ≤ λ max and || ¯ f k || L ∞ ( B ) ≤ M r k . Moreover, ¯ u k is a nonnegative minimizer in ¯ u k + W , ¯ p k ( · )0 ( B ) of the functional(3.22) ¯ J k ( v ) = Z B (cid:16) |∇ v | ¯ p k ( x ) ¯ p k ( x ) + ¯ λ k ( x ) χ { v> } + ¯ f k v (cid:17) dx with ¯ u k (0) = 0 and max B / ¯ u k ( x ) > k. Let d k ( x ) = dist( x, { ¯ u k = 0 } ) and O k = n x ∈ B : d k ( x ) ≤ − | x | o . Since ¯ u k (0) = 0 then B / ⊂ O k , therefore m k := sup O k (1 − | x | )¯ u k ( x ) ≥ max B / (1 − | x | )¯ u k ( x ) ≥
34 max B / ¯ u k ( x ) > k. For each fix k , ¯ u k is bounded, then (1 − | x | )¯ u k ( x ) → | x | → x k ∈ O k such that (1 − | x k | )¯ u k ( x k ) = sup O k (1 − | x | )¯ u k ( x ), and then(3.23) ¯ u k ( x k ) = m k − | x k | ≥ m k > k as x k ∈ O k , and δ k := d k ( x k ) ≤ −| x k | . Let y k ∈ ∂ { ¯ u k > } ∩ B such that | y k − x k | = δ k . Then,(1) B δ k ( y k ) ⊂ B , since if y ∈ B δ k ( y k ) ⇒ | y | < δ k + | x k | ≤ , (2) B δk ( y k ) ⊂ O k , since if y ∈ B δk ( y k ) ⇒ | y | ≤ δ k + | x k | ≤ − δ k ⇒ d k ( y ) ≤ δ k ≤ − | y | z ∈ B δk ( y k ) ⇒ − | z | ≥ − | x k | − | x k − z | ≥ − | x k | − δ k ≥ − | x k | . By (2) we havemax O k (1 − | x | )¯ u k ( x ) ≥ max B δk ( y k ) (1 − | x | )¯ u k ( x ) ≥ max B δk ( y k ) (1 − | x k | )2 ¯ u k ( x ) , NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 17 where in the last inequality we are using (3). Then,(3.24) 2¯ u k ( x k ) ≥ max B δk ( y k ) ¯ u k ( x ) . As B δ k ( x k ) ⊂ { ¯ u k > } then ∆ ¯ p k ( x ) ¯ u k = ¯ f k in B δ k ( x k ), and by Harnack’s inequality (Proposition3.1) we have(3.25) max B δk ( x k ) ¯ u k ( x ) ≤ C (cid:2) min B δk ( x k ) ¯ u k ( x ) + δ k (cid:3) , with C a positive constant depending only on N, p min , p max , L, M and M . We point out that, inorder to get this uniform constant C in (3.25), we have used, while applying Proposition 3.1, that γ k := sup B δk ( x k ) ¯ p k − inf B δk ( x k ) ¯ p k ≤ Lr k δ k ≤ Lr k , so that || ¯ u k || γ k L ∞ ( B δk ( x k )) ≤ ( M/r k ) Lr k ≤ C ( L, M ) . Recalling (3.23), we get from (3.25), for k large,(3.26) min B δk ( x k ) ¯ u k ( x ) ≥ c ¯ u k ( x k ) , with c a positive constant depending only on N, p min , p max , L, M and M . As B δ k ( x k ) ∩ B δk ( y k ) = ∅ we have by (3.26)(3.27) max B δk ( y k ) ¯ u k ( x ) ≥ c ¯ u k ( x k ) . Let w k ( x ) = ¯ u k ( y k + δ k x )¯ u k ( x k ) . Then, w k (0) = 0 and, by (3.24) and (3.27), we havemax B w k ≤ B / w k ≥ c > . (3.28)Now, recalling that ¯ u k is a nonnegative minimizer in ¯ u k + W , ¯ p k ( · )0 ( B ) of the functional ¯ J k in (3.22)and that B δk ( y k ) ⊂ B , we see that w k is a nonnegative minimizer of ˆ J k in w k + W , ¯ p k ( y k + δk x )0 ( B ),whereˆ J k ( v ) = Z B (cid:16) c ¯ p k ( y k + δk x ) k |∇ v | ¯ p k ( y k + δk x ) ¯ p k ( y k + δ k x ) + ¯ λ k ( y k + δ k x ) χ { v> } + ¯ f k ( y k + δ k x )¯ u k ( x k ) v (cid:17) dx, and c k = u k ( x k ) δ k .We now notice that c k → ∞ . So we define ˜ p k ( x ) = ¯ p k ( y k + δ k x ) and divide the functionalˆ J k by c ˜ p − k k , with ˜ p − k = inf B ˜ p k . Then, it follows that w k is a nonnegative minimizer of ˜ J k in w k + W , ˜ p k ( · )0 ( B ), where˜ J k ( v ) = Z B (cid:16) ˜ a k ( x ) |∇ v | ˜ p k ( x ) ˜ p k ( x ) + ˜ λ k ( x ) χ { v> } + ˜ f k v (cid:17) dx, ˜ a k ( x ) = c ˜ p k ( x ) − ˜ p − k k , ˜ λ k ( x ) = ¯ λ k ( y k + δ k x ) c − ˜ p − k k and ˜ f k ( x ) = ¯ f k ( y k + δ k x )¯ u k ( x k ) c − ˜ p − k k . We claim that(3.29) k ˜ f k k L ∞ ≤ ˜ M and ˜ f k → B , (3.30) ˜ λ k → B , (3.31) ˜ a k → ≤ ˜ a k ≤ M and k∇ ˜ a k k L ∞ ≤ L in B , (3.32) ˜ p k → p uniformly and p min ≤ p ≤ p max in B , up to a subsequence, for some constants ˜ M , M , L and p .In fact, (3.29) follows since | ˜ f k ( x ) | = | r k f k ( r k ( y k + δ k x )) u k ( r k x k ) r k c − ˜ p − k k | ≤ M M c − k →
0. On theother hand, 0 < ˜ λ k ( x ) ≤ λ max c − k → B there holds, for k large, that 1 ≤ ˜ a k ( x ) ≤ e k∇ ˜ p k k L ∞ log c k and k∇ ˜ a k k L ∞ ≤k∇ ˜ p k k L ∞ log c k k ˜ a k k L ∞ . But k∇ ˜ p k k L ∞ log c k ≤ Lr k δ k log (cid:0) Mr k δ k (cid:1) →
0, which implies (3.31).Finally, to see (3.32) we observe that p min ≤ p k ( x ) ≤ p max and k∇ p k k L ∞ (Ω) ≤ L and then, fora subsequence, p k → p uniformly on compacts of Ω, so ˜ p k ( x ) = p k ( r k ( y k + δ k x )) → p = p (0)uniformly in B .We now take v k the solution of(3.33) div (cid:0) ˜ a k ( x ) |∇ v k | ˜ p k ( x ) − ∇ v k (cid:1) = ˜ f k in B / , v k − w k ∈ W , ˜ p k ( · )0 ( B / ) . From Lemma 3.2, Remark 3.4 and the bounds in (3.28), (3.29) and (3.31), it follows that if k islarge enough(3.34) || v k || L ∞ ( B / ) ≤ ¯ C with ¯ C = ¯ C ( p min , ˜ M , L ) . Here we have used that k∇ ˜ p k k L ∞ ≤ Lr k δ k so k∇ ˜ p k k L ∞ / < p min − k large.Then, applying Theorem 1.1 in [14] we obtain that, for k large,(3.35) || v k || C ,α ( B / ) ≤ ˆ C with ˆ C = ˆ C ( p min , p max , ˜ M , L , L, M , N ) , for some 0 < α <
1. Therefore, there is a function v ∈ C ,α ( B / ) such that, for a subsequence,(3.36) v k → v and ∇ v k → ∇ v uniformly in B / . Moreover, (3.29), (3.31) and (3.32) imply that(3.37) ∆ p v = 0 in B / . Let us now show that(3.38) w k − v k → L p min ( B / ) . From the minimality of w k we have(3.39) Z B / ˜ a k ( x ) |∇ w k | ˜ p k ( x ) ˜ p k ( x ) − ˜ a k ( x ) |∇ v k | ˜ p k ( x ) ˜ p k ( x ) + Z B / ˜ f k ( w k − v k ) ≤ C ( N ) k ˜ λ k k L ∞ ( B / ) . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 19 Then, we can argue as in the proof of Theorem 3.2 and get estimate (3.10) for u = w k , v = v k , a ( x ) = ˜ a k ( x ), p ( x ) = ˜ p k ( x ), f = ˜ f k and r = 3 /
4, which together with (3.39), gives Z A k |∇ w k − ∇ v k | ˜ p k ( x ) dx ≤ C k ˜ λ k k L ∞ ( B / ) , (3.40) Z A k |∇ w k − ∇ v k | ( |∇ w k | + |∇ v k | ) ˜ p k ( x ) − dx ≤ C k ˜ λ k k L ∞ ( B / ) , (3.41)where A k = B / ∩ { ˜ p k ( x ) < } , A k = B / ∩ { ˜ p k ( x ) ≥ } and C = C ( p min , p max , N ).Applying H¨older’s inequality (Theorem A.3) with exponents p k ( x ) and − ˜ p k ( x ) , we get(3.42) Z A k |∇ w k − ∇ v k | ˜ p k ( x ) dx ≤ k F k k L / ˜ pk ( · ) ( A k ) k G k k L / (2 − ˜ pk ( · )) ( A k ) , where F k = |∇ w k − ∇ v k | ˜ p k ( |∇ w k | + |∇ v k | ) (˜ p k − p k / G k = ( |∇ w k | + |∇ v k | ) (2 − ˜ p k )˜ p k / . Since Z A k | F k | / ˜ p k ( x ) dx = Z A k |∇ w k − ∇ v k | ( |∇ w k | + |∇ v k | ) ˜ p k ( x ) − dx, then, from (3.41), (3.30) and Proposition A.1, we get, for k large,(3.43) k F k k L / ˜ pk ( · ) ( A k ) ≤ C k ˜ λ k k p min / L ∞ ( B / ) ,C = C ( p min , p max , N ). On the other hand, (3.33) and the bounds in (3.29), (3.31) and (3.34) give1 p max Z B / |∇ v k | ˜ p k ( x ) ≤ Z B / ˜ a k ( x ) |∇ v k | ˜ p k ( x ) ˜ p k ( x ) ≤ Z B / ˜ a k ( x ) |∇ w k | ˜ p k ( x ) ˜ p k ( x ) + Z B / ˜ f k ( w k − v k ) ≤ C (cid:0) Z B / |∇ w k | ˜ p k ( x ) (cid:1) . This implies(3.44) Z A k | G k | / (2 − ˜ p k ( x )) dx ≤ C Z B / ( |∇ w k | ˜ p k ( x ) + |∇ v k | ˜ p k ( x ) ) dx ≤ ˜ C (cid:0) Z B / |∇ w k | ˜ p k ( x ) (cid:1) , for some ˜ C = ˜ C ( p min , p max , ˜ M , M , L ) ≥
1. Now (3.44) and Proposition A.1 give(3.45) k G k k L / (2 − ˜ pk ( · )) ( A k ) ≤ ˜ C (cid:0) Z B / |∇ w k | ˜ p k ( x ) (cid:1) . Let us show that the right hand side in (3.45) can be bounded independently of k .In fact, let ˜ v k be the solution of(3.46) div (cid:0) ˜ a k ( x ) |∇ ˜ v k | ˜ p k ( x ) − ∇ ˜ v k (cid:1) = ˜ f k in B / , ˜ v k − w k ∈ W , ˜ p k ( · )0 ( B / ) . Then, similar arguments to those leading to (3.34) and (3.35), give, for k large enough,(3.47) || ˜ v k || L ∞ ( B / ) ≤ ¯ C with ¯ C = ¯ C ( p min , ˜ M , L ) , and(3.48) || ˜ v k || C ,α ( B / ) ≤ ˆ C with ˆ C = ˆ C ( p min , p max , ˜ M , L , L, M , N ) , for some 0 < α < w k is a nonnegative minimizer of ˜ J k in B , then we can argue as in the proof of Theorem3.2 and get estimate (3.15) for u = w k , v = ˜ v k , a ( x ) = ˜ a k ( x ), p ( x ) = ˜ p k ( x ), λ ( x ) = ˜ λ k ( x ), f = ˜ f k , r = 7 / ρ = 3 /
4. That is,(3.49) Z B / |∇ w k | ˜ p k ( x ) dx ≤ C + C Z B / |∇ ˜ v k | ˜ p k ( x ) dx, where C = C ( p min , p max , N, λ max ). Therefore (3.49) and (3.48) give, for k large, a uniform boundfor the right hand side in (3.45). That is,(3.50) k G k k L / (2 − ˜ pk ( · )) ( A k ) ≤ ¯ C, with ¯ C = ¯ C ( p min , p max , ˜ M , L , L, M , N, λ max ).Now, putting together (3.40), (3.42), (3.43), (3.50) and (3.30), we obtain(3.51) Z B / |∇ w k − ∇ v k | ˜ p k ( x ) → . Thus, using Poincare’s inequality (Theorem A.4 ) and Theorem A.2, we get (3.38).In order to conclude the proof, we now observe that, by Corollary 3.1, there exists 0 < γ < γ = γ ( N, p min ), such that k w k k C γ ( B / ) ≤ C with C = C ( p min , p max , ˜ M , L , L, M , N, λ max )(recall that k w k k L ∞ ( B ) ≤ w ∈ C γ ( B / ) such that, for a subsequence,(3.52) w k → w uniformly in B / . In addition, recalling (3.36), (3.37) and (3.38), we get v = w in B / and ∆ p w = 0 in B / .Finally, since there holds that w k ≥ w k (0) = 0 and (3.28), now (3.52) implies w ≥ , w (0) = 0 , max B / w ≥ c > , which contradicts the strong minimum principle and concludes the proof. (cid:3) We can now prove the Lipschitz continuity of nonnegative local minimizers
Corollary 3.2.
Let p, f, λ and u be as in Lemma 3.3. Then u is locally Lipschitz continuous in Ω . Moreover, for any Ω ′ ⊂⊂ Ω the Lipschitz constant of u in Ω ′ can be estimated by a constant C depending only on N , p min , p max , L , λ min , λ max , k u k L ∞ (Ω) , k f k L ∞ (Ω) and dist(Ω ′ , ∂ Ω) .Proof. The result is a consequence of Corollary 3.1, Lemma 3.3 and Theorem 3.3 above, andProposition 2.1 in [25]. (cid:3)
Next we have
Theorem 3.4.
Let p, f, λ and u be as in Lemma 3.3. Assume moreover that ∇ u ∈ L ∞ (Ω) . Thereexist positive constants c and ρ such that, for every x ∈ Ω ′ , u ( x ) ≥ c dist( x, { u ≡ } ) , if dist( x, { u ≡ } ) ≤ ρ. The constants depend only on p min , p max , L, || f || L ∞ (Ω) , λ min , λ max , ||∇ u || L ∞ (Ω) and dist(Ω ′ , ∂ Ω) . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 21 Proof.
We will prove the statement for x ∈ Ω ′ such that u ( x ) > u k ∈ W ,p k ( · ) (Ω) ∩ L ∞ (Ω) corresponding to functionals J k given by functions p k , f k and λ k , with p min ≤ p k ( x ) ≤ p max , k∇ p k k L ∞ ≤ L , || f k || L ∞ (Ω) ≤ L , λ min ≤ λ k ( x ) ≤ λ max , ||∇ u k || L ∞ (Ω) ≤ L and points x k ∈ Ω ′ , with u k ( x k ) >
0, such that d k = dist( x k , { u k ≡ } ) → u k ( x k ) d k → . Let us define in B , for d k small, w k ( x ) = d k u k ( x k + d k x ), ¯ p k ( x ) = p k ( x k + d k x ), ¯ f k ( x ) = d k f k ( x k + d k x ) and ¯ λ k ( x ) = λ k ( x k + d k x ). Then p min ≤ ¯ p k ( x ) ≤ p max , k∇ ¯ p k k L ∞ ( B ) ≤ Ld k , λ min ≤ ¯ λ k ( x ) ≤ λ max and || ¯ f k || L ∞ ( B ) ≤ L d k . Moreover, w k is a nonnegative local minimizer of the functional¯ J k ( v ) = Z B (cid:16) |∇ v | ¯ p k ( x ) ¯ p k ( x ) + ¯ λ k ( x ) χ { v> } + ¯ f k v (cid:17) dx. Since w k > B , we have ∆ ¯ p k ( x ) w k = ¯ f k in B (see (3.18)). In addition, w k (0) = u k ( x k ) d k → ||∇ w k || L ∞ ( B ) ≤ L . Then, by interior H¨older gradient estimates it follows that, for a subsequence, w k → w and ∇ w k → ∇ w uniformly on compact subsets of B . Moreover, for a subsequence,¯ f k → p k → p uniformly on compact subsets of B , with p constant. This implies that∆ p w = 0 in B .By Harnack’s inequality there exists a constant c >
0, depending on N and p , such thatsup B / w ≤ c inf B / w and therefore, given δ >
0, there exists k such that for k ≥ k sup B / w k ≤ c inf B / w k + C δ, for a constant C depending on N and p . In particular we have, for k large, w k ( x ) ≤ c w k (0) + C δ in B / . Let α k > u k ( x k ) = α k d k , this is, α k = w k (0). Let ψ ∈ C ∞ ( B ) such that ψ ≡ B / , ψ ≡ B \ B / , 0 ≤ ψ ≤ z k ( x ) = ( min (cid:16) w k ( x ) , ( cα k + C δ ) ψ (cid:17) in B / ,w k ( x ) outside B / . Then, z k ∈ W , ¯ p k ( · ) ( B ) and z k coincides with w k on ∂B so that there holds that ¯ J k ( z k ) ≥ ¯ J k ( w k ).Let D k = B / ∩ { w k > ( cα k + C δ ) ψ } . Observe that z k ≤ w k , so that χ { z k > } ≤ χ { w k > } . Inaddition, w k > B / , z k = 0 in B / and B / ⊂ D k . Therefore, if C δ ≤ and k is largeenough so that cα k ≤ , we get λ min | B / | ≤ Z D k ¯ λ k ( x ) (cid:8) χ { w k > } − χ { z k > } (cid:9) dx ≤ Z D k ( cα k + C δ ) p min p min |∇ ψ | ¯ p k + L d k Z D k (cid:2) ( cα k + C δ ) ψ + w k (cid:3) dx ≤ C ( cα k + C δ ) , with C = C ( ψ, p min , p max , L ). So that λ min | B / | ≤ C ( cα k + C δ ) , and, if CC δ ≤ λ min | B / | , it follows that12 λ min | B / | ≤ ¯ Cα k = ¯ C u k ( x k ) d k → , which is a contradiction. (cid:3) We also have
Lemma 3.4.
Let p and f be as in Theorem 3.1. Let Ω ′ ⊂⊂ Ω and u ∈ C (Ω) , u ≥ , ∇ u ∈ L ∞ (Ω) with ∆ p ( x ) u = f in { u > } be such that there exist positive constants c and ρ such that, for every x ∈ Ω ′ , there holds that u ( x ) ≥ c dist( x, { u ≡ } ) if dist( x, { u ≡ } ) ≤ ρ . Then, there exist positiveconstants δ and ρ such that for every x ∈ Ω ′ ∩ { u > } with d ( x ) = dist( x, { u ≡ } ) ≤ ρ , wehave sup B d ( x ) ( x ) u ≥ (1 + δ ) u ( x ) . The constants depend only on p min , p max , L, || f || L ∞ (Ω) , ||∇ u || L ∞ (Ω) , c , ρ and dist(Ω ′ , ∂ Ω) .Proof. Suppose by contradiction that there exist functions u k , p k , f k , with 1 < p min ≤ p k ( x ) ≤ p max < ∞ , k∇ p k k L ∞ ≤ L , || f k || L ∞ (Ω) ≤ L , u k ∈ C (Ω), u k ≥ ||∇ u k || L ∞ (Ω) ≤ L , with∆ p k ( x ) u k = f k in { u k > } and u k ( x ) ≥ c dist( x, { u k ≡ } ) if dist( x, { u k ≡ } ) ≤ ρ and x ∈ Ω ′ ,and sequences δ k → ρ k → x k ∈ Ω ′ ∩ { u k > } with d k = dist( x k , { u k ≡ } ) ≤ ρ k such thatsup B dk ( x k ) u k ≤ (1 + δ k ) u k ( x k ) . Take w k ( x ) = u k ( x k + d k x ) u k ( x k ) . Then, w k (0) = 1 andmax B w k ≤ (1 + δ k ) , w k > (cid:16)(cid:16) u k ( x k ) d k (cid:17) ¯ p k ( x ) − |∇ w k | ¯ p k ( x ) − ∇ w k (cid:17) = ¯ f k in B , where ¯ p k ( x ) = p k ( x k + d k x ) and ¯ f k ( x ) = d k f k ( x k + d k x ). On the other hand, we have c ≤ u k ( x k ) d k ≤ L , k∇ w k k L ∞ ( B ) ≤ L d k u k ( x k ) ≤ L c . Then, using the gradient estimates in [14], we deduce that, for a subsequence, u k ( x k ) d k → a ∈ [ c , L ], w k → w and ¯ p k → p ∈ R uniformly in B and ∇ w k → ∇ w uniformly on compact subsetsof B .There holds that ∆ p w = 0 in B , w (0) = 1 and w ≤ B . Therefore w ≡ B .Let y k ∈ ∂ { u k > } with | x k − y k | = d k . Then, if z k = y k − x k d k , we have w k ( z k ) = u k ( y k ) u k ( x k ) = 0and we may assume that z k → ¯ z ∈ ∂B . Thus, 1 = w (¯ z ) = 0. This is a contradiction, and thelemma is proved. (cid:3) As a consequence of the previous results, we obtain
Theorem 3.5.
Let p, f, λ and u be as in Theorem 3.4. Let Ω ′ ⊂⊂ Ω . There exist constants c > , r > such that if x ∈ Ω ′ ∩ ∂ { u > } and r ≤ r then sup B r ( x ) u ≥ cr. NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 23 The constants depend only on p min , p max , L, || f || L ∞ (Ω) , λ min , λ max , ||∇ u || L ∞ (Ω) and dist(Ω ′ , ∂ Ω) .Proof. We will follow the ideas of Theorem 1.9 in [8].Step 1. We will prove that there exist positive constants ¯ c , ¯ r and ¯ ρ such that if x ∈ Ω ′ ∩ { u > } ,dist( x , { u ≡ } ) ≤ ¯ ρ and r ≤ ¯ r , then sup B r ( x ) u ≥ ¯ cr. In fact, let ρ = dist(Ω ′ , ∂ Ω) and ˜Ω = B ρ / (Ω ′ ), so Ω ′ ⊂⊂ ˜Ω ⊂⊂ Ω.By Theorem 3.4 and Lemma 3.4 (applied to points in ˜Ω), there exist positive constants c and ρ such that, for every x ∈ ˜Ω with dist( x, { u ≡ } ) ≤ ρ , u ( x ) ≥ c dist( x, { u ≡ } ) , and positive constants δ and ρ such that for every x ∈ ˜Ω ∩ { u > } with d ( x ) = dist( x, { u ≡ } ) ≤ ρ , we have sup B d ( x ) ( x ) u ≥ (1 + δ ) u ( x ) . The constants depend only on p min , p max , L, || f || L ∞ (Ω) , ||∇ u || L ∞ (Ω) , dist( ˜Ω , ∂ Ω) = dist(Ω ′ , ∂ Ω), λ min and λ max .Let ¯ r = min { dist(Ω ′ , ∂ ˜Ω) , ρ, ρ } , ¯ ρ = ρ and r ≤ ¯ r . Let x ∈ Ω ′ ∩ { u > } such that d =dist( x , { u ≡ } ) ≤ ¯ ρ , then u ( x ) ≥ c d . There are two possibilities:i) d ≥ r .In this case u ( x ) ≥ c r and the result follows.ii) d < r .In this case, proceeding as in [8], we construct a polygonal that never leaves B r ( x ), startingat x and finishing at ˜ x ∈ B r ( x ), such that u (˜ x ) ≥ ˜ cr , with an explicit ˜ c > r and ¯ ρ as above, r ≤ ¯ r and x ∈ Ω ′ ∩ ∂ { u > } . We take x ∈ B r ( x ) ∩{ u > }∩ Ω ′ .Then, dist( x , { u ≡ } ) ≤ | x − x | ≤ ¯ ρ and thus, from the result in Step 1,sup B r ( x ) u ≥ sup B r ( x ) u ≥ ¯ c r . This completes the proof. (cid:3)
The following result in the section is
Theorem 3.6.
Let p, f, λ and u be as in Theorem 3.4. Let Ω ′ ⊂⊂ Ω . There exist constants ˜ c ∈ (0 , and ˜ r > such that, if x ∈ Ω ′ ∩ ∂ { u > } with B r ( x ) ⊂ Ω ′ and r ≤ ˜ r , there holds | B r ( x ) ∩ { u > }|| B r ( x ) | ≤ − ˜ c. The constants depend only on p min , p max , L, || f || L ∞ (Ω) , λ min , λ max , ||∇ u || L ∞ (Ω) and dist(Ω ′ , ∂ Ω) .Proof. Let us suppose by contradiction that there exist a sequence of nonnegative local minimizers u k ∈ W ,p k ( · ) (Ω) ∩ L ∞ (Ω) corresponding to functionals J k given by functions p k , f k and λ k , with p min ≤ p k ( x ) ≤ p max , k∇ p k k L ∞ ≤ L , || f k || L ∞ (Ω) ≤ L , λ min ≤ λ k ( x ) ≤ λ max , ||∇ u k || L ∞ (Ω) ≤ L and balls B r k ( x k ) ⊂ Ω ′ with x k ∈ ∂ { u k > } and r k →
0, such that | B r k ( x k ) ∩ { u k = 0 }|| B r k ( x k ) | → B rkσ ( x k ) u k ≥ cr k σ, for 0 < σ < , where c is the positive constant given by Theorem 3.5.Let ¯ u k ( x ) = u k ( x k + r k x ) r k , ¯ p k ( x ) = p k ( x k + r k x ) and ¯ f k ( x ) = r k f k ( x k + r k x ). Then p min ≤ ¯ p k ( x ) ≤ p max , k∇ ¯ p k k L ∞ ( B ) ≤ Lr k , || ¯ f k || L ∞ ( B ) ≤ L r k , 0 ∈ ∂ { ¯ u k > } , | B ∩ { ¯ u k = 0 }| = ε k → , (3.53) sup B σ ¯ u k ≥ cσ, for 0 < σ < , and ∆ ¯ p k ( x ) ¯ u k ≥ ¯ f k in B / . Let us take v k ∈ W , ¯ p k ( · ) ( B / ), such that(3.54) ∆ ¯ p k ( x ) v k = ¯ f k in B / , v k − ¯ u k ∈ W , ¯ p k ( · )0 ( B / ) . Observe that there holds that || ¯ u k || L ∞ ( B / ) ≤ L / || v k || L ∞ ( B / ) ≤ ¯ C with ¯ C = ¯ C ( L, p min , L , L ) , (this estimate follows from Lemma 3.2 and Remark 3.4, if k is large enough).Since v k ≥ ¯ u k then 0 ≤ χ { v k > } − χ { ¯ u k > } ≤ χ { ¯ u k =0 } and therefore, using that ¯ u k are nonnegativelocal minimizers, we get(3.56) Z B / (cid:16) |∇ ¯ u k | ¯ p k ( x ) ¯ p k ( x ) − |∇ v k | ¯ p k ( x ) ¯ p k ( x ) (cid:17) ≤ λ max | B ∩ { ¯ u k = 0 }| + L r k Z B / | ¯ u k − v k | . Applying (3.55), we now obtain(3.57) Z B / (cid:16) |∇ ¯ u k | ¯ p k ( x ) ¯ p k ( x ) − |∇ v k | ¯ p k ( x ) ¯ p k ( x ) (cid:17) ≤ C ( ε k + L r k ) . We claim that(3.58) Z B / |∇ ¯ u k − ∇ v k | ¯ p k ( x ) dx → . In fact, let u s ( x ) = s ¯ u k ( x ) + (1 − s ) v k ( x ). By using (3.54) and the inequalities in (1.4), we get Z B / |∇ ¯ u k | ¯ p k ( x ) ¯ p k ( x ) − |∇ v k | ¯ p k ( x ) ¯ p k ( x ) + Z B / ¯ f k (¯ u k − v k ) = Z dss Z B / (cid:16) |∇ u s | ¯ p k ( x ) − ∇ u s − |∇ v k | ¯ p k ( x ) − ∇ v k (cid:17) · ∇ ( u s − v k ) ≥ C (cid:16) Z B / ∩{ ¯ p k ≥ } |∇ ¯ u k − ∇ v k | ¯ p k ( x ) + Z B / ∩{ ¯ p k < } |∇ ¯ u k − ∇ v k | (cid:16) |∇ ¯ u k | + |∇ v k | (cid:17) ¯ p k ( x ) − (cid:17) . NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 25 Now (3.57) implies Z { ¯ p k ≥ }∩ B / |∇ ¯ u k − ∇ v k | ¯ p k ( x ) dx ≤ ˜ C ( ε k + L r k ) and Z { ¯ p k < }∩ B / (cid:0) |∇ ¯ u k | + |∇ v k | (cid:1) ¯ p k ( x ) − |∇ ¯ u k − ∇ v k | dx ≤ ˜ C ( ε k + L r k ) . From these inequalities we obtain, reasoning as in the proof of Theorem 5.1 in [5], Z B / |∇ ¯ u k − ∇ v k | ¯ p k ( x ) dx ≤ C max { ε k + L r k , ( ε k + L r k ) p min / } and thus, (3.58) follows.On the other hand, by interior H¨older gradient estimates, there holds that, for a subsequence, v k → v and ∇ v k → ∇ v uniformly on compact subsets of B / . Since k∇ ¯ p k k L ∞ ( B ) ≤ Lr k , thereexists a constant p such that (for a subsequence) ¯ p k → p uniformly in B / .Finally, since k∇ ¯ u k k L ∞ ( B / ) ≤ L we have, for a subsequence, ¯ u k → u uniformly in B / .Let w k = ¯ u k − v k . Then, w k → u − v uniformly on compact subsets of B / . By (3.58) we havethat k∇ w k k L ¯ pk ( · ) ( B / ) →
0. Since w k ∈ W , ¯ p k ( · )0 ( B / ), by Poincare’s inequality (Theorem A.4) weget that k w k k L ¯ pk ( · ) ( B / ) → || u − v || L p ( B / ) = 0. Thus, u = v .Now, using that v k → u locally in C ( B / ) and ¯ f k → B / , we deduce that∆ p u = ∆ p v = 0 in B / .As ¯ u k → u uniformly in B / we get, by (3.53), that sup B / u ≥ c . But u (0) = lim ¯ u k (0) =0 and u ≥
0. By the strong maximum principle we arrive at a contradiction and the resultfollows. (cid:3)
We devote the last part of the section to discuss the fulfillment of properties (3) and (4) in thedefinition of weak solution for nonnegative local minimizers of (1.1).We need
Definition 3.2.
Let p, f and λ be as in Definition 3.1 and let u ∈ W ,p ( · )+ δ (Ω), for some δ > D ⊂ Ω let J p,λ,fD ( v ) = J D ( v ) = Z D (cid:16) |∇ v | p ( x ) p ( x ) + λ ( x ) χ { v> } + f v (cid:17) dx. We say that u is a mild minimizer of J in Ω if for every B r ( x ) ⊂⊂ Ω and v ∈ W ,p ( · )+ δ ( B r ( x ))with v − u ∈ W ,p ( · )+ δ ( B r ( x )), for some 0 < δ < δ , J B r ( x ) ( u ) ≤ J B r ( x ) ( v ) . We have the following results for mild minimizers
Proposition 3.2.
Let p, f and λ be as in Theorem 3.1. Assume moreover that λ ∈ C (Ω) . Let u be a nonnegative Lipschitz mild minimizer of J in Ω . Let x k ∈ Ω ∩ ∂ { u > } , x k → x ∈ Ω , ρ k → and u k ( x ) = u ( x k + ρ k x ) ρ k . Assume that u k → u uniformly on compact sets of R N . Then u is a nonnegative Lipschitz mild minimizer of J in R N , with p ( x ) ≡ p ( x ) , λ ( x ) ≡ λ ( x ) and f ≡ . Proof.
Let B r = B r (¯ x ) be any ball in R N and assume for simplicity that ¯ x = 0. Denote p k ( x ) = p ( x k + ρ k x ), p = p ( x ), λ k ( x ) = λ ( x k + ρ k x ), λ = λ ( x ), f k ( x ) = ρ k f ( x k + ρ k x ) and J r,k ( v ) = Z B r (cid:16) |∇ v | p k ( x ) p k ( x ) + λ k ( x ) χ { v> } + f k v (cid:17) dx,J r, ( v ) = Z B r (cid:16) |∇ v | p p + λ χ { v> } (cid:17) dx. Let v ∈ W ,p + δ ( B r ) with v − u ∈ W ,p + δ ( B r ) for some δ >
0. We want to show that(3.59) J r, ( u ) ≤ J r, ( v ) . For h > v h,k = ( v in B r ,u + | x |− rh ( u k − u ) in B r + h \ B r . Then, since p k ≤ p + δ/ B r + h for k large, it follows that v h,k ∈ W ,p k ( · )+ δ/ ( B r + h ), v h,k − u k ∈ W ,p k ( · )+ δ/ ( B r + h ), for k large, and there holds J r + h,k ( v h,k ) = Z B r + h (cid:16) |∇ v h,k | p k ( x ) p k ( x ) + λ k ( x ) χ { v h,k > } + f k v h,k (cid:17) = J r, ( v ) + Z B r + h \ B r (cid:16) |∇ v h,k | p k ( x ) p k ( x ) + λ k ( x ) χ { v h,k > } + f k v h,k (cid:17) + Z B r (cid:16) |∇ v | p k ( x ) p k ( x ) − |∇ v | p p + ( λ k ( x ) − λ ) χ { v> } + f k v (cid:17) ≤ J r, ( v ) + C hr N − + C Z B r + h \ B r | u k − u | p k ( x ) h p k ( x ) + Z B r (cid:16) |∇ v | p k ( x ) p k ( x ) − |∇ v | p p + ( λ k ( x ) − λ ) χ { v> } + f k v (cid:17) . Therefore,(3.60) lim sup k →∞ J r + h,k ( v h,k ) ≤ J r, ( v ) + C hr N − . On the other hand, λ χ { u > } ≤ lim inf k →∞ λ k ( x ) χ { u k > } , which implies(3.61) Z B r λ χ { u > } dx ≤ lim inf k →∞ Z B r λ k ( x ) χ { u k > } dx. In addition, since ∇ u k ⇀ ∇ u weakly in L p ( B r ), arguing in a similar way as in Theorem 3.1,we get(3.62) Z B r |∇ u | p p dx ≤ lim inf k →∞ Z B r |∇ u k | p p dx = lim inf k →∞ Z B r |∇ u k | p k ( x ) p k ( x ) dx. Now, using (3.61) and (3.62), and the fact that u k are nonnegative Lipschitz mild minimizers of J with p ( x ) = p k ( x ), λ ( x ) = λ k ( x ) and f ( x ) = f k ( x ) we obtain J r, ( u ) ≤ lim inf k →∞ J r,k ( u k ) ≤ lim inf k →∞ J r + h,k ( u k ) + C hr N − ≤ lim inf k →∞ J r + h,k ( v h,k ) + C hr N − , NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 27 which in combination with (3.60) gives J r, ( u ) ≤ J r, ( v ) + C hr N − . Therefore, letting h → (cid:3) We will need
Proposition 3.3.
Let < p and λ , α be positive constants. Let u be a Lipschitz mild minimizerof J in R N , with p ( x ) ≡ p , λ ( x ) ≡ λ and f ≡ . Assume that u = αx +1 in B r , for some r > .Then, α = (cid:16) p p − λ (cid:17) /p .Proof. Let ε > τ ε ( x ) = x + εφ ( | x | ) e with φ ∈ C ∞ ( − r , r ), and let u ε ( x ) = u ( τ ε − ( x )).Then, u ε ∈ W ,p + δ ( B r ) with u ε − u ∈ W ,p + δ ( B r ), for some δ >
0, which implies that0 ≤ J r , ( u ε ) − J r , ( u ) , for J r , ( v ) = Z B r (cid:16) |∇ v | p p + λ χ { v> } (cid:17) dx. We now proceed as in Lemma 7.3 in [27]. In fact, there it is proved an analogous result with J r , replaced by J ( v ) = Z B r (cid:16) G ( |∇ v | ) + λχ { v> } (cid:17) dx, for a general G and a positive constant λ , and it is shown that(3.63) G ′ ( α ) α − G ( α ) = λ. Since in our case we have J with G ( t ) = t p p and λ = λ , [27] applies and thus (3.63) yields α p − α p p = λ , which gives the desired result. (cid:3) Next we prove
Theorem 3.7.
Let p, f, λ and u be as in Lemma 3.3. Assume moreover that λ ∈ C (Ω) . Let x ∈ Ω ∩ ∂ { u > } . Then, lim sup x → x u ( x ) > |∇ u ( x ) | = λ ∗ ( x ) , where λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) .Proof. Let α := lim sup x → x u ( x ) > |∇ u ( x ) | . Since u ∈ Lip loc (Ω), 0 ≤ α < ∞ . By the definition of α there exists a sequence z k → x such that u ( z k ) > , |∇ u ( z k ) | → α. Let y k be the nearest point from z k to Ω ∩ ∂ { u > } and let d k = | z k − y k | .Consider the blow up sequence u d k with respect to B d k ( y k ). That is, u d k ( x ) = d k u ( y k + d k x ).Since u is locally Lipschitz, and u d k (0) = 0 for every k , there exists u , with u (0) = 0, such that (for a subsequence) u d k → u uniformly on compact sets of R N . Moreover, using Lemma 3.3 andinterior H¨older estimates we deduce that ∇ u d k → ∇ u uniformly on compact subsets of { u > } .We claim that |∇ u | ≤ α in R N . In fact, let R > δ >
0. Then, there exists τ > |∇ u ( x ) | ≤ α + δ for any x ∈ B τ R ( x ). For | z k − x | < τ R/ d k < τ / B d k R ( z k ) ⊂ B τ R ( x ) and therefore, |∇ u d k ( x ) | ≤ α + δ in B R − for k large. Passing to the limit,we obtain |∇ u | ≤ α + δ in B R − , and since δ and R were arbitrary, the claim holds.Now, if α = 0, since u (0) = 0, it follows that u ≡
0. This contradicts Theorem 3.5 and then, α > γ >
0, ( u ) γ ( x ) = γ u ( γx ). There exist a sequence γ n → u ∈ Lip ( R N )such that ( u ) γ n → u uniformly on compact sets of R N .Using Lemma 3.3 and Theorem 3.6 and proceeding as in the proof of Theorem 5.1 in [24] weobtain that u ( x ) = αx +1 .Now, since u is a nonnegative local minimizer of functional J in Ω, then u is locally Lipschitzand it is a nonnegative mild minimizer of J in Ω. Thus, applying Proposition 3.2 to u and to theblow up sequence u d k , we get that u is a nonnegative Lipschitz mild minimizer of J in R N , with p ( x ) ≡ p ( x ), λ ( x ) ≡ λ ( x ) and f ≡ u and to the blow up sequence ( u ) γ n , we alsoget that u ( x ) = αx +1 is a nonnegative Lipschitz mild minimizer of J in R N , with p ( x ) ≡ p ( x ), λ ( x ) ≡ λ ( x ) and f ≡ α = λ ∗ ( x ). (cid:3) Our next result is
Theorem 3.8.
Let p, f, λ and u be as in Theorem 3.7. Let x ∈ Ω ∩ ∂ { u > } . Assume there is aball B contained in { u = 0 } touching x , then (3.64) lim sup x → x u ( x ) > u ( x ) dist ( x, B ) = λ ∗ ( x ) , where λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) .Proof. Let ℓ be the finite limit on the left hand side of (3.64) and let y k → x with u ( y k ) > u ( y k ) d k → ℓ, d k = dist( y k , B ) . Consider the blow up sequence u k with respect to B d k ( x k ), where x k ∈ ∂B are points with | x k − y k | = d k , that is, u k ( x ) = u ( x k + d k x ) d k . Choose a subsequence with blow up limit u , such that there exists e := lim k →∞ y k − x k d k . Using Lemma 3.3 and Theorem 3.5 and proceeding as in the proof of Theorem 5.2 in [24] wehave that u ( x ) = ℓ h x, e i + . Thus, applying Propositions 3.2 and 3.3, we get that ℓ = λ ∗ ( x ). (cid:3) The last result in this section is
Theorem 3.9.
Let p, f, λ and u be as in Theorem 3.7. Let x ∈ Ω ∩ ∂ { u > } be such that ∂ { u > } has at x an inward unit normal ν in the measure theoretic sense. Then, u ( x ) = λ ∗ ( x ) h x − x , ν i + + o ( | x − x | ) , NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 29 where λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) .Proof. Take u λ ( x ) = λ u ( x + λx ) . Let ρ > B ρ ( x ) ⊂⊂ Ω. Since u λ ∈ Lip ( B ρ/λ )uniformly in λ , u λ (0) = 0, there exist λ j → U such that u λ j → U uniformly on compactsets of R N . Since |∇ u ( x ) | ≤ L in B r ( x ) for some positive L and r then, for any M > |∇ u λ j ( x ) | ≤ L in B M (0) for j large. Therefore, |∇ U ( x ) | ≤ L in R N and U ∈ Lip ( R N ).Without loss of generality we assume that x = 0, and ν = e . From Lemma 3.3, ∆ p ( λx ) u λ = λf ( λx ) in { u λ > } . Using the fact that e is the inward normal in the measure theoretic sense,we have, for fixed k , |{ u λ > } ∩ { x < } ∩ B k | → λ → . Hence, U = 0 in { x < } . Moreover, U is nonnegative in { x > } , ∆ p U = 0 in { U > } with p = p ( x ) and U vanishes in { x ≤ } . Then, by Lemma A.1 we have that there exists α ≥ U ( x ) = αx +1 + o ( | x | ) . Define U λ ( x ) = λ U ( λx ), then U λ → αx +1 uniformly on compact sets of R N .Now, by Theorem 3.5 and Remark 2.1, we have, for some c > < r < r ,1 r N Z B r u λ j dx ≥ cr and then 1 r N Z B r U λ j dx ≥ cr. Therefore α >
0. Now, since u is a nonnegative local minimizer of functional J in Ω, then u islocally Lipschitz and it is a nonnegative mild minimizer of J in Ω. Thus, by Proposition 3.2, U isa nonnegative Lipschitz mild minimizer of J in R N with p ( x ) ≡ p ( x ), λ ( x ) ≡ λ ( x ) and f ≡ U we get that U = αx +1 is also a nonnegative Lipschitz mildminimizer of J in R N with p ( x ) ≡ p ( x ), λ ( x ) ≡ λ ( x ) and f ≡ α = λ ∗ ( x ) . We have shown that U ( x ) = ( λ ∗ ( x ) x + o ( | x | ) x > x ≤ . Then, using that ∆ p ( λx ) u λ = λf ( λx ) in { u λ > } , by interior H¨older gradient estimates we have ∇ u λ j → ∇ U uniformly on compact subsets of { U > } . Then, by Theorem 3.7, |∇ U | ≤ λ ∗ ( x ) in R N . As U = 0 on { x = 0 } we have, U ≤ λ ∗ ( x ) x in { x > } .Now, proceeding as in the proof of Theorem 5.3 in [24], we conclude that U ≡ λ ∗ ( x ) x +1 and theresult follows. (cid:3) Energy minimizers of energy functional (1.2)In this section we prove existence of minimizers of the energy functional (1.2) and, in the spiritof the previous section, we develop an exhaustive analysis of the essential properties of functions u ε which are nonnegative local minimizers of that energy. As a consequence we obtain results forsolutions u ε to the singular perturbation problem P ε ( f ε , p ε ) which are nonnegative local energyminimizers and moreover, we get results for their limit functions u .We start by pointing out that the same considerations in Definition 3.1 and Remarks 3.1 and3.2 for functional (1.1) apply to functional (1.2) in the present section. We first obtain
Theorem 4.1.
Let Ω ⊂ R N be a bounded domain and let φ ε ∈ W ,p ε ( · ) (Ω) be such that k φ ε k ,p ε ( · ) ≤A , with < p min ≤ p ε ( x ) ≤ p max < ∞ and k∇ p ε k L ∞ ≤ L . Let f ε ∈ L ∞ (Ω) such that k f ε k L ∞ (Ω) ≤A . There exists u ε ∈ W ,p ε ( · ) (Ω) that minimizes the energy (4.1) J ε ( v ) = Z Ω (cid:16) |∇ v | p ε ( x ) p ε ( x ) + B ε ( v ) + f ε v (cid:17) dx among functions v ∈ W ,p ε ( · ) (Ω) such that v − φ ε ∈ W ,p ε ( · )0 (Ω) . Here B ε ( s ) = R s β ε ( τ ) dτ .Then, the function u ε satisfies (4.2) ∆ p ε ( x ) u ε = β ε ( u ε ) + f ε in Ω and for every Ω ′ ⊂⊂ Ω there exists C = C (Ω ′ , A , A , p min , p max , L ) such that (4.3) sup Ω ′ u ε ≤ C. Proof.
Let us prove first that a minimizer exists. In fact, let K ε = n v ∈ W ,p ε ( · ) (Ω) : v − φ ε ∈ W ,p ε ( · )0 (Ω) o . In order to prove that J ε is bounded from below in K ε , we observe that if v ∈ K ε , then J ε ( v ) ≥ p max Z Ω |∇ v | p ε ( x ) + Z Ω f ε v dx, and we have, by Theorem A.3 and Theorem A.4, Z Ω | f ε v | dx ≤ k f ε k p ε ′ ( · ) k v k p ε ( · ) ≤ k f ε k p ε ′ ( · ) ( k v − φ ε k p ε ( · ) + k φ ε k p ε ( · ) ) ≤ C k∇ v − ∇ φ ε k p ε ( · ) + C ≤ C k∇ v k p ε ( · ) + C . If (cid:16) R Ω |∇ v | p ε ( x ) dx (cid:17) /p min ≥ (cid:16) R Ω |∇ v | p ε ( x ) dx (cid:17) /p max we get, by Proposition A.1, Z Ω | f ε v | dx ≤ C (cid:16) Z Ω |∇ v | p ε ( x ) dx (cid:17) /p min + C ≤ C + 12 p max Z Ω |∇ v | p ε ( x ) dx. If, on the other hand, (cid:16) R Ω |∇ v | p ε ( x ) dx (cid:17) /p min < (cid:16) R Ω |∇ v | p ε ( x ) dx (cid:17) /p max , we get in an analogousway Z Ω | f ε v | dx ≤ C (cid:16) Z Ω |∇ v | p ε ( x ) dx (cid:17) /p max + C ≤ C + 12 p max Z Ω |∇ v | p ε ( x ) dx. Taking C = max { C , C } , we get(4.4) J ε ( v ) ≥ − C + 12 p max Z Ω |∇ v | p ε ( x ) dx ≥ − C , which shows that J ε is bounded from below in K ε .At this point we want to remark that the constants C , ..., C above can be taken depending onlyon A , A , p min , p max and L .We now take a minimizing sequence { u n } ⊂ K ε . Without loss of generality we can assume that J ε ( u n ) ≤ J ε ( φ ε ), so by (4.4), R Ω |∇ u n | p ε ( x ) ≤ C . By Proposition A.1, k∇ u n − ∇ φ ε k p ε ( · ) ≤ C and, NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 31 as u n − φ ε ∈ W ,p ε ( · )0 (Ω), by Theorem A.4 we have k u n − φ ε k p ε ( · ) ≤ C . Therefore, by Theorem A.1there exist a subsequence (that we still call u n ) and a function u ε ∈ W ,p ε ( · ) (Ω) such that(4.5) || u ε || W ,pε ( · ) (Ω) ≤ ¯ C, with ¯ C = ¯ C ( A , A , p min , p max , L ) ,u n ⇀ u ε weakly in W ,p ε ( · ) (Ω) , and, by Theorem A.2, u n ⇀ u ε weakly in W ,p min (Ω) . Now, by the compactness of the immersion W ,p min (Ω) ֒ → L p min (Ω) we have that, for a subsequencethat we still denote by u n , u n → u ε in L p min (Ω) ,u n → u ε a.e. Ω . As K ε is convex and closed, it is weakly closed, so u ε ∈ K ε .It follows that lim n →∞ Z Ω B ε ( u n ) dx = Z Ω B ε ( u ε ) dx, lim n →∞ Z Ω f ε u n dx = Z Ω f ε u ε dx, Z Ω |∇ u ε | p ε ( x ) p ε ( x ) dx ≤ lim inf n →∞ Z Ω |∇ u n | p ε ( x ) p ε ( x ) dx. In order to prove the last inequality we proceed as in (3.4) in Theorem 3.1.Hence J ε ( u ε ) ≤ lim inf n →∞ J ε ( u n ) = inf v ∈K ε J ε ( v ) . Therefore, u ε is a minimizer of J ε in K ε .Let us now prove that there holds (4.2). Let t > ξ ∈ C ∞ (Ω). Using the minimality of u ε we have0 ≤ t ( J ε ( u ε − tξ ) − J ε ( u ε )) = 1 t Z Ω (cid:16) |∇ u ε − t ∇ ξ | p ε ( x ) p ε ( x ) − |∇ u ε | p ε ( x ) p ε ( x ) (cid:17) dx +1 t Z Ω (cid:16) B ε ( u ε − tξ ) − B ε ( u ε ) (cid:17) dx + 1 t Z Ω (cid:16) f ε ( u ε − tξ ) − f ε u ε (cid:17) dx ≤ − Z Ω |∇ u ε − t ∇ ξ | p ε ( x ) − ( ∇ u ε − t ∇ ξ ) · ∇ ξ dx + 1 t Z Ω (cid:16) B ε ( u ε − tξ ) − B ε ( u ε ) (cid:17) dx − Z Ω f ε ξ dx and if we take t →
0, we obtain(4.6) 0 ≤ − Z Ω |∇ u ε | p ε ( x ) − ∇ u ε · ∇ ξ dx − Z Ω β ε ( u ε ) ξ dx − Z Ω f ε ξ dx. If we now take t <
0, and proceed in a similar way, we obtain the opposite sign in (4.6) and (4.2)follows.Finally, in order to prove (4.3), we observe that, from Proposition A.1 and estimate (4.5), wehave that R Ω | u ε | p ε ( x ) dx ≤ ¯ C ( A , A , p min , p max , L ). Thus, the desired estimate follows from theapplication of Proposition 2.1 in [35], since ∆ p ε ( x ) u ε ≥ f ε ≥ −A in Ω. (cid:3) Remark 4.1.
We are interested in studying the behavior of a family u ε of nonnegative localminimizers of the energy J ε defined in (4.1).If u ε are as in Theorem 4.1 then u ε satisfy (4.2) and it follows from Proposition 2.1 in [35] that u ε ∈ L ∞ loc (Ω) . Moreover, by Theorem 1.1 in [14] u ε ∈ C (Ω) and ∇ u ε are locally H¨older continuousin Ω.If we have, for instance, that φ ε ≥ f ε ≤ u ε ≥ ε > ξ ε = min( u ε , ∈ W ,p ε ( · )0 (Ω). Then, we get (4.6)for the test function ξ ε and, using that β ε ( u ε ) ξ ε = 0 and f ε ≤
0, we obtain R Ω |∇ ξ ε | p ε ( x ) dx = 0,which implies u ε ≥ Remark 4.2.
Let u ε be a family of nonnegative local minimizers of the energy functional J ε definedin (4.1) which are uniformly bounded, with f ε and p ε uniformly bounded (like for instance the oneconstructed in Theorem 4.1 and Remark 4.1). Then, as in Theorem 4.1 we deduce that u ε aresolutions to P ε ( f ε , p ε ) and thus, all the results in our work [24] apply to this family. In particular,there hold the local uniform gradient estimates of Theorem 2.1 in [24] and the results on passageto the limit in Lemma 3.1 in [24].We also have Theorem 4.2.
Assume that < p min ≤ p ε j ( x ) ≤ p max < ∞ and that k∇ p ε j k L ∞ ≤ L . Let u ε j ∈ W ,p εj ( · ) (Ω) be nonnegative local minimizers of (4.7) J ε j ( v ) = Z Ω (cid:16) |∇ v | p εj ( x ) p ε j ( x ) + B ε j ( v ) + f ε j v (cid:17) dx, with k u ε j k L ∞ (Ω) ≤ L and k f ε j k L ∞ (Ω) ≤ L , such that u ε j → u uniformly on compact subsets of Ω , f ε j ⇀ f ∗− weakly in L ∞ (Ω) , p ε j → p uniformly on compact subsets of Ω and ε j → . Then, u islocally Lipschitz. Let B r = B r ( x ) ⊂⊂ Ω and denote J ( v ) = Z Ω (cid:16) |∇ v | p ( x ) p ( x ) + M χ { v> } + f v (cid:17) dx, (4.8) J r, ( v ) = Z B r (cid:16) |∇ v | p ( x ) p ( x ) + M χ { v> } + f v (cid:17) dx, (4.9) where M = R β ( s ) ds .i) If v ∈ W ,p ( · )+ δ ( B r ) for some δ > and v − u ∈ W ,p ( · )0 ( B r ) , then J r, ( u ) ≤ J r, ( v ) .ii) If there holds that p ε j ≤ p in Ω and u ∈ W ,p ( · ) (Ω) , then u is a nonnegative local minimizer offunctional (4.8) .Proof. We first observe that the estimates of Theorem 2.1 in [24] apply, as well as the results inLemma 3.1 in [24]. In particular, u ε j are locally uniformly Lipschitz and therefore u is locallyLipschitz in Ω.We will follow the ideas in Theorem 1.16 in [8]. In fact, let B r = B r ( x ) ⊂⊂ Ω, for simplicityassume x = 0, and denote J r,j ( v ) = Z B r (cid:16) |∇ v | p εj ( x ) p ε j ( x ) + B ε j ( v ) + f ε j v (cid:17) dx,J r, ( v ) = Z B r (cid:16) |∇ v | p ( x ) p ( x ) + M χ { v> } + f v (cid:17) dx. NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 33 Let us first assume that ii) holds.Given v ∈ W ,p ( · ) ( B r ) such that v − u ∈ W ,p ( · )0 ( B r ), we want to show that(4.10) J r, ( u ) ≤ J r, ( v ) . For h > v h,j = ( v in B r ,u + | x |− rh ( u ε j − u ) in B r + h \ B r . Then, since p ε j ≤ p , it follows that v h,j ∈ W ,p εj ( · ) ( B r + h ), v h,j − u ε j ∈ W ,p εj ( · )0 ( B r + h ) and thereholds J r + h,j ( v h,j ) = Z B r + h (cid:16) |∇ v h,j | p εj ( x ) p ε j ( x ) + B ε j ( v h,j ) + f ε j v h,j (cid:17) ≤ J r, ( v )+ Z B r + h \ B r (cid:16) |∇ v h,j | p εj ( x ) p ε j ( x ) + B ε j ( v h,j ) + f ε j v h,j (cid:17) + Z B r (cid:16) |∇ v | p εj ( x ) p ε j ( x ) − |∇ v | p ( x ) p ( x ) + ( f ε j − f ) v (cid:17) ≤ J r, ( v ) + C hr N − + C Z B r + h \ B r | u ε j − u | p εj ( x ) h p εj ( x ) + Z B r (cid:16) |∇ v | p εj ( x ) p ε j ( x ) − |∇ v | p ( x ) p ( x ) + ( f ε j − f ) v (cid:17) . Therefore,(4.11) lim sup j →∞ J r + h,j ( v h,j ) ≤ J r, ( v ) + C hr N − . On the other hand,
M χ { u> } ≤ lim inf j →∞ B ε j ( u ε j ) , which implies(4.12) Z B r M χ { u> } dx ≤ lim inf j →∞ Z B r B ε j ( u ε j ) dx. In addition, since ∇ u ε j ⇀ ∇ u weakly in L p ( · ) ( B r ), arguing in a similar way as in Theorem 4.1, weget(4.13) Z B r |∇ u | p ( x ) p ( x ) dx ≤ lim inf j →∞ Z B r |∇ u ε j | p ( x ) p ( x ) dx = lim inf j →∞ Z B r |∇ u ε j | p εj ( x ) p ε j ( x ) dx. Now, using (4.12) and (4.13), and the fact that u ε j are nonnegative local minimizers of J ε j , weobtain J r, ( u ) ≤ lim inf j →∞ J r,j ( u ε j ) ≤ lim inf j →∞ J r + h,j ( u ε j ) + C hr N − ≤ lim inf j →∞ J r + h,j ( v h,j ) + C hr N − , which in combination with (4.11) gives J r, ( u ) ≤ J r, ( v ) + C hr N − . Therefore, letting h → v h,j ∈ W ,p εj ( · ) ( B r + h ), v h,j − u ε j ∈ W ,p εj ( · )0 ( B r + h ) for large j . (cid:3) Remark 4.3.
Let u ε be a family of nonnegative local minimizers of J ε ( v ) = R Ω (cid:0) |∇ v | pε ( x ) p ε ( x ) + B ε ( v ) + f ε v (cid:1) dx , with 1 < p min ≤ p ε ( x ) ≤ p max < ∞ , k∇ p ε k L ∞ ≤ L , k u ε k L ∞ (Ω) ≤ L and k f ε k L ∞ (Ω) ≤ L .Then, with a minor modification of the proof of Theorem 3.4, we can prove that, given Ω ′ ⊂⊂ Ω,there exist positive constants c and ρ such that, for every x ∈ Ω ′ , u ε > ε in B d ( x ) with 0 < d ≤ ρ, implies u ε ( x ) ≥ c d , and, in particular, u ε ( x ) ≥ c dist( x , { u ε ≤ ε } ) , if dist( x , { u ε ≤ ε } ) ≤ ρ, with c and ρ depending only on p min , p max , L, L , L , M = R β ( s ) ds and dist(Ω ′ , ∂ Ω).As a consequence it follows that, if u = lim u ε j as ε j → x ∈ Ω ′ , u ( x ) ≥ c dist( x , { u ≡ } ) , if dist( x , { u ≡ } ) ≤ ρ. As in the case of minimizers of the energy (1.1), for minimizers of the singular perturbationproblem we have
Theorem 4.3.
Let p ε j , f ε j , u ε j , ε j , p , f and u be as in Theorem 4.2. Let Ω ′ ⊂⊂ Ω . There existconstants c > , r > such that if x ∈ Ω ′ ∩ ∂ { u > } and r ≤ r then sup B r ( x ) u ≥ cr. The constants depend only on N, p min , p max , L, L , L , M, || β || L ∞ and dist(Ω ′ , ∂ Ω) .Proof. The proof follows as that of Theorem 3.5, replacing Theorem 3.4 by Remark 4.3. (cid:3)
In an analogous way as we obtained for minimizers of functional (1.1), for minimizers of thesingular perturbation problem we have
Theorem 4.4.
Let p ε j , f ε j , u ε j , ε j , p , f and u be as in Theorem 4.2. Let Ω ′ ⊂⊂ Ω . There existconstants ˜ c ∈ (0 , and ˜ r > such that, if x ∈ Ω ′ ∩ ∂ { u > } with B r ( x ) ⊂ Ω ′ and r ≤ ˜ r , thereholds | B r ( x ) ∩ { u > }|| B r ( x ) | ≤ − ˜ c. The constants depend only on N, p min , p max , L, L , L , M, || β || L ∞ and dist(Ω ′ , ∂ Ω) .Proof. The proof follows as that of Theorem 3.6. In this case we obtain estimate (3.56) by usingpart i) in Theorem 4.2, since v k ∈ W , ¯ p k ( · )+ δ k ( B / ), for some δ k > (cid:3) Regularity of the Free Boundary
In this section, we first consider nonnegative local minimizers to the energy functional (1.1) andwe obtain results on the regularity of the free boundary for these functions, which are a consequenceof the results in Section 3 and the results in our work [25].In addition, we consider any family u ε of nonnegative local minimizers to the energy functional(1.2) which are uniformly bounded, with f ε and p ε uniformly bounded (like, for instance, the oneconstructed in Theorem 4.1 and Remark 4.1). Then (recall Remark 4.2), all the results in ourprevious paper [24] apply to such a family. Hence, as a consequence of the results in Section 4 andin our work [25], we obtain results on the regularity of the free boundary for limit functions of thisfamily.First, for nonnegative local minimizers to the energy functional (1.1), we get NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 35 Theorem 5.1.
Assume that < p min ≤ p ( x ) ≤ p max < ∞ with k∇ p k L ∞ ≤ L , f ∈ L ∞ (Ω) and < λ min ≤ λ ( x ) ≤ λ max < ∞ with λ ∈ C (Ω) . Let u ∈ W ,p ( · ) (Ω) ∩ L ∞ (Ω) be a nonnegative localminimizer of (1.1) in a domain Ω ⊂ R N .Then, u is a weak solution to the free boundary problem: u ≥ and ( P ( f, p, λ ∗ )) ( ∆ p ( x ) u = f in { u > } u = 0 , |∇ u | = λ ∗ ( x ) on ∂ { u > } with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − λ ( x ) (cid:17) /p ( x ) .Proof. The result follows by applying Lemma 3.3, Corollary 3.2 and Theorems 3.3, 3.5, 3.7, 3.8and 3.9. (cid:3)
Now, we can apply the results in [25] and deduce
Theorem 5.2.
Let p , f , λ and u be as in Theorem 5.1. Assume moreover that f ∈ W ,q (Ω) , p ∈ W ,q (Ω) with q > max { , N/ } and λ is H¨older continuous in Ω .Then, there is a subset R of the free boundary Ω ∩ ∂ { u > } ( R = ∂ red { u > } ) which is locallya C ,α surface, for some < α < , and the free boundary condition is satisfied in the classicalsense in a neighborhood of R . Moreover, R is open and dense in Ω ∩ ∂ { u > } and the remainderof the free boundary has ( N − − dimensional Hausdorff measure zero.If moreover ∇ p and f are H¨older continuous in Ω , then the equation is satisfied in the classicalsense in a neighborhood of R .Proof. We first observe that, by Theorem 5.1, Theorem 4.4 in [25] applies at every x ∈ Ω ∩ ∂ red { u > } .Finally we observe that, since u is a weak solution to P ( f, p, λ ∗ ), Theorem 2.1 in [25] and Lemma2.3 in [25] apply to u . Therefore, recalling Theorem 3.6 we deduce, from Theorem 4.5.6(3) in [15],that H N − ( ∂ { u > } \ ∂ red { u > } ) = 0. (cid:3) We also obtain higher regularity from the application of Corollary 4.1 in [25]
Corollary 5.1.
Let p , f , λ and u be as in Theorem 5.2. Assume moreover that p ∈ C (Ω) , f ∈ C (Ω) and λ ∈ C (Ω) then ∂ red { u > } ∈ C ,µ for every < µ < .If p ∈ C m +1 ,µ (Ω) , f ∈ C m,µ (Ω) and λ ∈ C m +1 ,µ (Ω) for some < µ < and m ≥ , then ∂ red { u > } ∈ C m +2 ,µ .Finally, if p , f and λ are analytic, then ∂ red { u > } is analytic. Next, for minimizers of the energy functional (1.2) we obtain, as a consequence of the results inSection 4 and the results in [24]
Theorem 5.3.
Assume that < p min ≤ p ε j ( x ) ≤ p max < ∞ and k∇ p ε j k L ∞ ≤ L . Let u ε j ∈ W ,p εj ( · ) (Ω) be a family of nonnegative local minimizers of (4.7) in a domain Ω ⊂ R N such that u ε j → u uniformly on compact subsets of Ω , f ε j ⇀ f ∗− weakly in L ∞ (Ω) , p ε j → p uniformly oncompact subsets of Ω and ε j → .Then, u is a weak solution to the free boundary problem: u ≥ and ( P ( f, p, λ ∗ )) ( ∆ p ( x ) u = f in { u > } u = 0 , |∇ u | = λ ∗ ( x ) on ∂ { u > } with λ ∗ ( x ) = (cid:16) p ( x ) p ( x ) − M (cid:17) /p ( x ) and M = R β ( s ) ds . Proof.
The result follows by applying first Remark 4.2 and Theorems 4.3 and 4.4 and then, Theorem6.1 in [24]. (cid:3)
We can now apply the results in [25] and deduce
Theorem 5.4.
Let p ε j , f ε j , u ε j , ε j , p , f and u be as in Theorem 5.3. Assume moreover that f ∈ W ,q (Ω) and p ∈ W ,q (Ω) with q > max { , N/ } .Then, there is a subset R of the free boundary Ω ∩ ∂ { u > } ( R = ∂ red { u > } ) which is locallya C ,α surface, for some < α < , and the free boundary condition is satisfied in the classicalsense in a neighborhood of R . Moreover, R is open and dense in Ω ∩ ∂ { u > } and the remainderof the free boundary has ( N − − dimensional Hausdorff measure zero.If moreover ∇ p and f are H¨older continuous in Ω , then the equation is satisfied in the classicalsense in a neighborhood of R .Proof. We first observe that, by Theorem 5.3, Theorem 4.4 in [25] applies at every x ∈ Ω ∩ ∂ red { u > } .Finally we observe that, since u is a weak solution to P ( f, p, λ ∗ ), Theorem 2.1 in [25] and Lemma2.3 in [25] apply to u . Therefore, recalling Theorem 4.4 we deduce, from Theorem 4.5.6(3) in [15],that H N − ( ∂ { u > } \ ∂ red { u > } ) = 0. (cid:3) We also obtain higher regularity from the application of Corollary 4.1 in [25]
Corollary 5.2.
Let p , f and u be as in Theorem 5.4. Assume moreover that p ∈ C (Ω) and f ∈ C (Ω) , then ∂ red { u > } ∈ C ,µ for every < µ < .If p ∈ C m +1 ,µ (Ω) and f ∈ C m,µ (Ω) for some < µ < and m ≥ , then ∂ red { u > } ∈ C m +2 ,µ .Finally, if p and f are analytic, then ∂ red { u > } is analytic. Appendix
A.In Section 1 we included some preliminaries on Lebesgue and Sobolev spaces with variableexponent. For the sake of completeness we collect here some additional results on these spaces aswell as some other results that are used throughout the paper.
Proposition A.1.
There holds min n(cid:16) Z Ω | u | p ( x ) dx (cid:17) /p min , (cid:16) Z Ω | u | p ( x ) dx (cid:17) /p max o ≤ k u k L p ( · ) (Ω) ≤ max n(cid:16) Z Ω | u | p ( x ) dx (cid:17) /p min , (cid:16) Z Ω | u | p ( x ) dx (cid:17) /p max o . Some important results for these spaces are
Theorem A.1.
Let p ′ ( x ) such that p ( x ) + 1 p ′ ( x ) = 1 . Then L p ′ ( · ) (Ω) is the dual of L p ( · ) (Ω) . Moreover, if p min > , L p ( · ) (Ω) and W ,p ( · ) (Ω) are reflexive. Theorem A.2.
Let q ( x ) ≤ p ( x ) . If Ω has finite measure, then L p ( · ) (Ω) ֒ → L q ( · ) (Ω) continuously. We also have the following H¨older’s inequality
NHOMOGENEOUS MINIMIZATION PROBLEMS FOR THE p ( x )-LAPLACIAN 37 Theorem A.3.
Let p ′ ( x ) be as in Theorem A.1. Then there holds Z Ω | f || g | dx ≤ k f k p ( · ) k g k p ′ ( · ) , for all f ∈ L p ( · ) (Ω) and g ∈ L p ′ ( · ) (Ω) . The following version of Poincare’s inequality holds
Theorem A.4.
Let Ω be bounded. Assume that p ( x ) is log-H¨older continuous in Ω (that is, p hasa modulus of continuity ω ( r ) = C (log r ) − ). For every u ∈ W ,p ( · )0 (Ω) , the inequality k u k L p ( · ) (Ω) ≤ C k∇ u k L p ( · ) (Ω) holds with a constant C depending only on N, diam(Ω) and the log-H¨older modulus of continuityof p ( x ) . For the proof of these results and more about these spaces, see [13], [18], [31], [17] and thereferences therein.We will also need
Lemma A.1.
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IMAS - CONICET and Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, (1428) Buenos Aires, Argentina.
E-mail address , Claudia Lederman: [email protected]
E-mail address , Noemi Wolanski:, Noemi Wolanski: