Almost minimizers for certain fractional variational problems
aa r X i v : . [ m a t h . A P ] M a y ALMOST MINIMIZERS FOR CERTAIN FRACTIONALVARIATIONAL PROBLEMS
SEONGMIN JEON AND ARSHAK PETROSYAN
To Nina Nikolaevna Uraltseva on the occasion of her 85th birthday.
Abstract.
In this paper we introduce a notion of almost minimizers for cer-tain variational problems governed by the fractional Laplacian, with the helpof the Caffarelli-Silvestre extension. In particular, we study almost fractionalharmonic functions and almost minimizers for the fractional obstacle problemwith zero obstacle. We show that for a certain range of parameters, almostminimizers are almost Lipschitz or C ,β -regular. Introduction and Main Results
Fractional harmonic functions.
Given 0 < s <
1, we say that a function u ∈ L s ( R n ) := L ( R n , (1 + | x | n +2 s ) − ) is s -fractional harmonic in an open setΩ ⊂ R n if(1.1) ( − ∆ x ) s u ( x ) := C n,s p.v. Z R n u ( x ) − u ( x + z ) | z | n +2 s = 0 in Ω , where p.v. stands for Cauchy’s principal value and C n,s is a normalization constant.The formula above is just one of many equivalent definitions of the fractional Lapla-cian ( − ∆ x ) s , another one being a pseudo-differential operator with Fourier symbol | ξ | s . We refer to a recent review of Garofalo [Gar19] for basic properties of ( − ∆ x ) s ,as well as many historical remarks concerning that operator.In recent years, there has been a surge of interest in nonlocal problems involvingthe fractional Laplacian, when it was discovered that the problems can be localizedby the use of the so-called Caffarelli-Silvestre extension procedure [CS07]. Namely,for a = 1 − s ∈ ( − , P ( x, y ) := C n,a | y | − a ( | x | + | y | ) n +1 − a , ( x, y ) ∈ R n × R + = R n +1+ , (to be called the Poisson kernel for the extension operator L a ) and consider theconvolution, still denoted by u , u ( x, y ) := u ∗ P ( · , y ) = Z R n u ( z ) P ( x − z, y ) dz, ( x, y ) ∈ R n +1+ . Mathematics Subject Classification.
Primary 49N60, 35R35.
Key words and phrases.
Almost minimizers, fractional Laplacian, fractional harmonic func-tions, fractional obstacle problem, regularity of solutions .The second author is supported in part by NSF Grant DMS-1800527.
Note that u ( x, y ) solves the Cauchy problem L a u := div( | y | a ∇ u ) = 0 in R n +1+ ,u ( x,
0) = u ( x ) on R n , where ∇ = ∇ x,y is the full gradient in x and y variables. L a is known as theCaffarelli-Silvestre extension operator . Then, one can recover ( − ∆ x ) s u as the frac-tional normal derivative on R n ( − ∆ x ) s u ( x ) = − C n,a lim y → y a ∂ y u ( x, y ) , x ∈ R n to be understood in the appropriate sense of traces. Now, going back to the def-inition (1.1), if we consider the even reflection of u in y -variable to all of R n +1 ,i.e., u ( x, y ) = u ( x, − y ) , x ∈ R n , y < , then the following fact holds: u ( x ) is s -fractional harmonic in Ω if and only if u ( x, y )satisfies(1.2) L a u = 0 in e Ω := R n +1 − ∪ (Ω × { } ) ∪ R n +1+ . (We will refer to solutions of L a u = 0 as L a -harmonic functions .) This is essentiallyLemma 4.1 in [CS07]. Since L a u = 0 in R n ± by definition, the condition (1.2) isequivalent to asking L a u = 0 in B r ( x ) , for any ball B r ( x ) centered at x ∈ Ω such that B r ( x ) ⋐ e Ω, or equivalently B ′ r ( x ) ⋐ Ω. Now, observing that the solutions of the above equation are minimizersof the weighted Dirichlet energy R B r ( x ) |∇ v | | y | a , we obtain the following fact. Proposition 1.1.
A function u ∈ L s ( R n ) is s -fractional harmonic in Ω if andonly if its reflected Caffarelli-Silvestre extension u ( x, y ) is in W , ( e Ω , | y | a ) and forany ball B r ( x ) with x ∈ Ω such that B ′ r ( x ) ⋐ Ω , we have Z B r ( x ) |∇ u | | y | a ≤ Z B r ( x ) |∇ v | | y | a , for any v ∈ u + W , ( B r ( x ) , | y | a ) . We take this proposition as the starting point for the definition of almost s -fractional harmonic functions, in the spirit of Anzellotti [Anz83]. Definition 1.2 (Almost s -fractional harmonic functions) . Let r > ω :(0 , r ) → [0 , ∞ ) be a modulus of continuity . We say that a function u ∈ L s ( R n )is almost s -fractional harmonic in an open set Ω ⊂ R n , with a gauge function ω ,if its reflected Caffarelli-Silvestre extension u ( x, y ) is in W , ( e Ω , | y | a ) and for anyball B r ( x ) with x ∈ Ω and 0 < r < r such that B ′ r ( x ) ⋐ Ω, we have(1.3) Z B r ( x ) |∇ u | | y | a ≤ (1 + ω ( r )) Z B r ( x ) |∇ v | | y | a , for any v ∈ u + W , ( B r ( x ) , | y | a ). i.e., a nondecreasing function with ω (0+) = 0 LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 3
Fractional obstacle problem.
A function u ∈ L s ( R n ) is said to solve the s -fractional obstacle problem with obstacle ψ in an open set Ω ⊂ R n , if(1.4) min { ( − ∆ x ) s u, u − ψ } = 0 in Ω . We refer to [Sil07, CSS08] for general introduction and basic results on this problem.With the help of the reflected Caffarelli-Silvestre extension, we can rewrite theproblem as a Signorini-type problem for the operator L a : L a u = 0 in R n +1 ± min {− ∂ ay u, u − ψ } = 0 in Ω , where ∂ ay u ( x,
0) := lim y → y a ∂ y u ( x, y ) . This, in turn, can be written in the following variational form, see [CSS08].
Proposition 1.3.
A function u ∈ L s ( R n ) solves (1.4) if and only if its reflectedCaffarelli-Silvestre extension u ( x, y ) is in W , ( e Ω) and for any ball B r ( x ) with x ∈ Ω such that B ′ r ( x ) ⋐ Ω , we have Z B r ( x ) |∇ u | | y | a ≤ Z B r ( x ) |∇ v | | y | a , for any v ∈ K ψ,u ( B r ( x ) , | y | a ) := { v ∈ u + W , ( B r , | y | a ) : v ≥ ψ on B ′ r ( x ) } . Definition 1.4 (Almost minimizers for s -fractional obstacle problem) . Let r > ω : (0 , r ) → [0 , ∞ ) be a modulus of continuity. We say that a function u ∈ L s ( R n ) is an almost minimizer for the s -fractional obstacle problem in an openset Ω ⊂ R n , with a gauge function ω , if its reflected Caffarelli-Silvestre extension u ( x, y ) is in W , ( e Ω , | y | a ) and for any ball B r ( x ) with x ∈ Ω and 0 < r < r suchthat B ′ r ( x ) ⋐ Ω, we have(1.5) Z B r ( x ) |∇ u | | y | a ≤ (1 + ω ( r )) Z B r ( x ) |∇ v | | y | a , for any v ∈ K ψ,u ( B r ( x ) , | y | a ).The notion of almost minimizers above is related to the one for the thin obstacleproblem ( s = 1 /
2) studied by the authors in [JP19], but there are certain importantdifferences. In Definition 1.4, we ask the almost minimizing property (1.5) to holdonly for balls centered on the “thin space” R n , while in [JP19], we ask that propertyfor balls centered at any point in an open set in the “thick space” R n +1 . In a sense,this means that here we think of the perturbation from minimizers as living on thethin space, while in [JP19] they live in the thick space.1.3. Main results and structure of the paper.
In this paper, our main concernis the regularity of almost minimizers in their original variables.We start with examples of almost minimizers in Section 2. We then proceed toprove the following results, echoing those in [Anz83] and [JP19].
Theorem I.
Let u ∈ L s ( R n ) be almost s -fractional harmonic in Ω . Then (1) u is almost Lipschitz in Ω , i.e, u ∈ C ,σ (Ω) for any < σ < . (2) If ω ( r ) = r α , then u ∈ C ,β (Ω) for some β = β n,a,α > . (3) If < s < / or s = 1 / and ω ( r ) = r α for some α > , then u is actually s -fractional harmonic in Ω . SEONGMIN JEON AND ARSHAK PETROSYAN
In the case of the s -fractional obstacle problem, our results are obtained underthe assumption that 1 / ≤ s < ψ = 0. Theorem II.
Let u ∈ L s ( R n ) be an almost minimizer for the s -fractional obstacleproblem with obstacle ψ = 0 in Ω . (1) If / ≤ s < , then u ∈ C ,σ (Ω) for any < σ < . (2) If / ≤ s < and ω ( r ) = r α for some α > , then u ∈ C ,β (Ω) for some β = β n,a,α > . The proofs follow the general approach in [Anz83] and [JP19] by first obtaininggrowth estimates for minimizers (see Section 3) and then deriving their perturbedversions for almost minimizers (Section 4 for s -fractional harmonic functions andSection 5 for the s -fractional obstacle problem). The regularity then follows byan embedding theorem of a Morrey-Campanato-type space into the H¨older space,which we included in Appendix A. Finally, Appendix B contains the proof of orthog-onal polynomial expansion of L a -harmonic functions, that we rely on in derivingthe growth estimates in Section 3. The polynomial expansion has other interestingcorollaries such as the (known) real-analyticity of s -fractional harmonic functions,which are of independent interest.1.4. Notation.
Throughout the paper we use the following notation. R n is the n -dimensional Euclidean space. The points of R n +1 are denoted by X = ( x, y ), where x = ( x , . . . , x n ) ∈ R n , y ∈ R . We routinely identify x ∈ R n with ( x, ∈ R n × { } . R n +1 ± stands for open halfspaces { X = ( x, y ) ∈ R n +1 : ± y > } .We use the following notations for balls of radius r in R n and R n +1 B r ( X ) = { Z ∈ R n +1 : | X − Z | < r } , (Euclidean) ball in R n +1 ,B ± r ( x ) = B r ( x, ∩ {± y > } , half-ball in R n +1 ,B ′ r ( x ) = B r ( x, ∩ { y = 0 } , ball in R n . We typically drop the center from the notation if it is the origin. Thus, B r = B r (0), B ′ r = B ′ r (0), etc.Next, ∇ u = ∇ X u = ( ∂ x u, . . . , ∂ x n u, ∂ y u ) stands for the full gradient, while ∇ x u = ( ∂ x u, . . . , ∂ x n u ). We also use the standard notations for partial derivatives,such as ∂ x i u , u x i , u y etc.In integrals, we often drop the variable and the measure of integration if it iswith respect to the Lebesgue measure or the surface measure. Thus, Z B r u | y | a = Z B r u ( X ) | y | a dX, Z ∂B r u | y | a = Z ∂B r u ( X ) | y | a dS X , where S X stands for the surface measure.By L ( B R , | y | a ) and L ( ∂B R , | y | a ) we indicate the weighted Lebesgue spaces offunctions with the norms k u k L ( B R , | y | a ) = Z B R u | y | a k u k L ( ∂B R , | y | a ) = Z ∂B R u | y | a .W , ( B R , | y | a ) is the corresponding weighted Sobolev space of functions with thenorm k u k W , ( B R , | y | a ) = k u k L ( B R , | y | a ) + k∇ u k L ( B R , | y | a ) . LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 5
We also use other typical notations for Sobolev spaces. Thus, W , ( B R , | y | a ) standsfor the closure of C ∞ ( B R ) in W , ( B R , | y | a ).For x ∈ R n and r >
0, we indicate by h u i x,r the | y | a -weighted integral meanvalue of a function u over B r ( x ). That is, h u i x,r = − Z B r ( x ) u | y | a = 1 ω n +1+ a r n +1+ a Z B r ( x ) u | y | a , where ω n +1+ a = R B | y | a is the | y | a -weighted volume of the unit ball B in R n +1 .Similarly to the other notations, we drop the origin if it is 0 and write h u i r for h u i ,r . 2. Examples of almost minimizers
Before we proceed with the proofs of the main results, we would like to give someexamples of almost minimizers.
Example 2.1.
Let u ∈ L s ( R n ) be a solution of( − ∆ x ) s u + b ( x ) · ∇ x u = 0 in Ω , where b = ( b , b , . . . , b n ) ∈ W , ∞ (Ω) and 1 / < s < − < a < u is an almost s -fractional harmonic with a gauge function ω ( r ) = Cr − a (note that − a > Proof.
Consider a ball B r ( x ) centered at x ∈ Ω such that B ′ r ( x ) ⋐ Ω. Withoutloss of generality assume that x = 0. Let v be the minimizer of Z B r |∇ v | | y | a on u + W , ( B r , | y | a ). Then Z B r ∇ v ∇ ( u − v ) | y | a = 0 , and as a consequence, Z B r ( |∇ u | − |∇ v | ) | y | a = Z B r |∇ ( u − v ) | | y | a . Then, we have Z B r ( |∇ u | − |∇ v | ) | y | a = 2 Z B + r |∇ ( u − v ) | | y | a = 2 Z B + r |∇ ( u − v ) | | y | a + div( | y | a ∇ ( u − v )) ( u − v )= 2 Z B + r div (cid:18) | y | a ∇ (cid:18) ( u − v ) (cid:19)(cid:19) = 2 Z ( ∂B r ) + | y | a ( u − v )( u ν − v ν ) − Z B ′ r ( u − v )( ∂ ay u − ∂ ay v )= C Z B ′ r ( u − v )( − ∆ x ) s u = − C Z B ′ r ( u − v ) b i u x i SEONGMIN JEON AND ARSHAK PETROSYAN with C = C n,a . Next, extending b i to R n +1 by b i ( x, y ) := b i ( x ), we have Z B r ( |∇ u | − |∇ v | ) | y | a = − C Z B ′ r ( u − v ) b i u x i = C Z B + r ∂ y (cid:0) ( u − v ) b i u x i (cid:1) = C Z B + r ( u y − v y ) b i u x i + ( u − v ) b i u x i y ≤ C k b k W , ∞ (Ω) Z B + r |∇ u | + |∇ v | + C Z ∂ ( B + r ) ( u − v ) b i u y ν x i − C Z B + r ∂ x i (( u − v ) b i ) u y = C k b k W , ∞ (Ω) Z B + r |∇ u | + |∇ v | − C Z B + r (( u x i − v x i ) b i + ( u − v ) b ix i ) u y ≤ C k b k W , ∞ (Ω) Z B + r |∇ u | + |∇ v | + | u − v | . Using Poincare’s inequality, it follows that Z B r | y | a ( |∇ u | − |∇ v | ) ≤ C Z B r |∇ u | + |∇ v | ≤ Cr − a Z B r ( |∇ u | + |∇ v | ) | y | a ≤ Cr − a Z B r |∇ u | | y | a . Hence, Z B r ( x ) |∇ u | | y | a ≤ (1 + Cr − a ) Z B r ( x ) |∇ v | | y | a , for 0 < r < r , with C and r depending on n , a , and k b k W , ∞ (Ω) . (cid:3) Example 2.2.
Let u ∈ L s ( R n ) be a solution of the obstacle problem for fractionalLaplacian with drift min { ( − ∆ x ) s u + b ( x ) · ∇ x u, u } = 0 in Ω , where b = ( b , b , . . . , b n ) ∈ W , ∞ (Ω) and 1 / < s < − < a < u isan almost minimizer for s -fractional obstacle problem in Ω with an obstacle ψ = 0and a gauge function ω ( r ) = Cr − a .The obstacle problem above has been studied earlier in [PP15] and [GPPS17]. Proof.
We argue similarly to Example 2.1. Let B r ( x ) centered at x ∈ Ω suchthat B ′ r ( x ) ⋐ Ω. Without loss of generality assume that x = 0. Let v be theminimizer of Z B r |∇ v | | y | a LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 7 on K ,u ( B r , | y | a ) = { v ∈ u + W , ( B r , | y | a ) : v ≥ B ′ r ( x ) } . Next, we write Z B r ( |∇ u | − |∇ v | ) | y | a = 2 Z B r ∇ u ∇ ( u − v ) | y | a − Z B r |∇ ( u − v ) | | y | a ≤ Z B r ∇ u ∇ ( u − v ) | y | a = 4 Z B + r ∇ u ∇ ( u − v ) | y | a + div( | y | a ∇ u )( u − v )= − Z B ′ r ( u − v ) ∂ ay u = C Z B ′ r ( u − v )( − ∆ x ) s u = C (cid:20) − Z B ′ r ∩{ u> } ( u − v ) b i u x i + Z B ′ r ∩{ u =0 } ( − v ) ( − ∆ x ) s u (cid:21) ≤ C (cid:20) − Z B ′ r ∩{ u> } ( u − v ) b i u x i − Z B ′ r ∩{ u =0 } ( − v ) b i u x i (cid:21) = − C Z B ′ r ( u − v ) b i u x i , where we used that ( − ∆) s u + b i u x i ≥ − v ≤ B ′ r ∩ { u = 0 } in the lastinequality.Then we complete the proof as in Example 2.1. (cid:3) Growth estimates for minimizers
In this section we prove growth estimates for L a -harmonic functions and solutionsof the Signorini problem for L a , i.e., minimizers of v of the weighted Dirichletintegral Z B r |∇ v | | y | a on v + W , ( B r , | y | a ) or on the thin obstacle constraint set K ,v ( B r , | y | a ).The idea is that these estimates will extend to almost minimizers and will ul-timately imply their regularity with the help of Morrey-Campanato-type spaceembedding.The proofs in this section are akin to those in [JP19] for almost minimizers ofthe thin obstacle problem. Yet, one has to be careful with different growth ratesfor tangential and normal derivatives.3.1. Growth estimates for L a -harmonic functions.Lemma 3.1. Let v ∈ W , ( B R , | y | a ) be a solution of L a v = 0 in B R . If v is evenin y , then for < ρ < R Z B ρ |∇ x v | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ x v | | y | a Z B ρ | v y | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R | v y | | y | a . SEONGMIN JEON AND ARSHAK PETROSYAN
Proof.
Note that we can write v ( x, y ) = ∞ X k =0 p k ( x, y ) , where p k ’s are L a -harmonic homogeneous polynomials of degree k (see Appendix B).Then { ∂ x i p k } ∞ k =1 are L a -harmonic homogeneous polynomials of degree k −
1, andthus orthogonal in L ( ∂B , | y | a ). Thus, Z B ρ |∇ x v | | y | a = ∞ X k =1 Z B ρ |∇ x p k | | y | a = ∞ X k =1 (cid:16) ρR (cid:17) n +1+ a +2( k − Z B R |∇ x p k | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a ∞ X k =1 Z B R |∇ x p k | | y | a = (cid:16) ρR (cid:17) n +1+ a Z B R |∇ x v | | y | a . Similarly, {| y | a ∂ y p k } ∞ k =1 are L − a -harmonic homogeneous functions of degree k − a , and thus orthogonal in L ( ∂B , | y | − a ). Notice that since p ( x, y ) = p ( x ) isindependent of y variable by the even symmetry, we have | y | a ∂ y p = 0. Thus, Z B ρ | v y | | y | a = Z B ρ || y | a v y | | y | − a = ∞ X k =2 Z B ρ || y | a ∂ y p k | | y | − a = ∞ X k =2 (cid:16) ρR (cid:17) n +1 − a +2( k − a ) Z B R || y | a ∂ y p k | | y | − a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R | v y | | y | a . (cid:3) Lemma 3.2.
Let v be a solution of L a v = 0 in B R , even in y . Then, for < ρ < R , (3.1) Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − h∇ x v i R | | y | a . Proof.
First note that since L a ( ∇ x v ) = 0 in B R , h∇ x v i = ∇ x v (0) by the meanvalue theorem for L a -harmonic functions, see [CSS08, Lemma 2.9]. If we use theexpansion v = P ∞ k =0 p k ( x, y ) in B R as in the proof of Lemma 3.1, then ∇ x v −∇ x v (0) = P ∞ k =2 ∇ x p k and consequently Z B ρ |∇ x v − ∇ x v (0) | | y | a = ∞ X k =2 Z B ρ |∇ x p k | | y | a = ∞ X k =2 (cid:16) ρR (cid:17) n + a +2 k − Z B R |∇ x p k | | y | a LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 9 ≤ (cid:16) ρR (cid:17) n + a +3 ∞ X k =2 Z B R |∇ x p k | | y | a = (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − ∇ x v (0) | | y | a . (cid:3) Growth estimates for the solutions of the Signorini problem for L a . Our estimates for the solutions of the Signorini problem will require an assumptionthat 1 / ≤ s <
1, or a ≤
0. Also, unless stated otherwise, the obstacle ψ is assumedto be zero.The first estimate is the analogue of Lemma 3.1, but with less information ofthe growth of v y . Lemma 3.3.
Let v be a solution of the Signorini problem for L a in B R , even in y ,with a ≤ . Then, for < ρ < R (3.2) Z B ρ |∇ v | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ v | | y | a . Proof.
We use the following property: if v is as in the statement of the lemma,then v x i , i = 1 , . . . , n , and y | y | a − v y are H¨older continuous in B R , see [CSS08].Moreover, we have that L a ( v ± x i ) ≥ , L − a (( y | y | a − v y ) ± ) ≥ B R . This follows from the fact that L a v x i = 0 in {± v x i > } and L − a ( y | y | a − v y ) = 0 in {± y | y | a − v y > } , by the complementarity condition v y v = 0 on B ′ R , as well as anargument in Exercise 2.6 or Exercise 9.5 in [PSU12]. As a consequence, we have L a ( |∇ x v | ) ≥ , L − a ( || y | a v y | ) ≥ B R . We next use the following | y | a -weighted sub-mean value property for L a -subharmonicfunctions: If L a w ≥ B R , − < a <
1, then ρ ρ n +1+ a Z B ρ w | y | a is nondecreasing. This follows by integration from the spherical sub-mean valueproperty, see [CSS08, Lemma 2.9]. Thus, we have that ρ ρ n +1+ a Z B ρ |∇ x v | | y | a ρ ρ n +1 − a Z B ρ | y | a u y are monotone nondecreasing for 0 < ρ < R . This implies Z B ρ |∇ x v | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ x v | | y | a Z B ρ v y | y | a ≤ (cid:16) ρR (cid:17) n +1 − a Z B R v y | y | a . In the case a ≤
0, we therefore conclude that the bound (3.2) holds. (cid:3)
Lemma 3.4.
Let v be a solution of the Signorini problem for L a in B R , even in y , with a ≤ . If v (0) = 0 , then there exists C = C n,α such that for < ρ < r < (3 / R , Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:16) ρr (cid:17) n + a +3 Z B r |∇ x v − h∇ x v i r | | y | a + C k v k L ∞ ( B R ) ρ n +2 R s Proof.
Define ϕ ( r ) := 1 r n + a +3 Z B r |∇ x v − h∇ x v i r | | y | a . Then, ϕ ( r ) = 1 r n + a +3 (cid:20)Z B r |∇ x v | | y | a − h∇ x v i r Z B r ∇ x v | y | a + h∇ x v i r Z B r | y | a (cid:21) = 1 r n + a +3 "Z B r |∇ x v | | y | a − ω n +1+ a r n +1+ a (cid:18)Z B r ∇ x v | y | a (cid:19) . Thus, using the Cauchy-Schwarz and Young’s inequality, we obtain ϕ ′ ( r ) = 1 r n + a +3 (cid:20) − n + a + 3 r Z B r |∇ x v | | y | a + Z ∂B r |∇ x v | | y | a + n + a + 3 ω n +1+ a r n +2+ a (cid:18)Z B r ∇ x v | y | a (cid:19) + n + 1 + aω n +1+ a r n +2+ a (cid:18)Z B r ∇ x v | y | a (cid:19) − ω n +1+ a r n +1+ a (cid:18)Z B r ∇ x v | y | a (cid:19) (cid:18)Z ∂B r ∇ x v | y | a (cid:19) (cid:21) ≥ − Cr n + a +3 " r Z B r |∇ x v | | y | a + (cid:18) r Z B r |∇ x v | | y | a (cid:19) / (cid:18)Z ∂B r |∇ x v | | y | a (cid:19) / ≥ − Cr n + a +3 (cid:20) r Z B r |∇ x v | | y | a + Z ∂B r |∇ x v | | y | a (cid:21) . Next, we note that [ ∇ x v ] C ,s ( B / R ) ≤ C n,s R s k v k L ∞ ( B R ) . Indeed, this follows from the known interior regularity for solutions of the Signoriniproblem for L a in B in the case R = 1, see e.g. [CSS08], and a simple scalingargument for all R >
0. Noting also that ∇ x v (0) = 0, since v attains its minimumon B ′ r at 0, we have that for X ∈ B r with r < (3 / R |∇ x v ( X ) | = |∇ x v ( X ) − ∇ x v (0) | ≤ CR s k v k L ∞ ( B R ) r s and so 1 r Z B r |∇ x v | | y | a + Z ∂B r |∇ x v | | y | a ≤ C k v k L ∞ ( B R ) r n +1 R s . This gives ϕ ′ ( r ) ≥ − Cr a +2 k v k L ∞ ( B R ) R s . Thus, for 0 < ρ < r < (3 / R , ϕ ( r ) − ϕ ( ρ ) = Z rρ ϕ ′ ( t ) dt ≥ − C k v k L ∞ ( B R ) ρ − − a − r − − a R s . LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 11
Therefore, Z B ρ |∇ x v − h∇ x v i ρ | | y | a = ρ n + a +3 ϕ ( ρ ) ≤ ρ n + a +3 (cid:18) ϕ ( r ) + C k v k L ∞ ( B R ) ρ − − a − r − − a R s (cid:19) ≤ (cid:16) ρr (cid:17) n + a +3 Z B r |∇ x v − h∇ x v i r | | y | a + C k v k L ∞ ( B R ) ρ n +2 R s . (cid:3) Lemma 3.5.
Let v be a solution of the Signorini problem for L a in B R , even in y .Then there are C = C n,a , C = C n,a such that for all < ρ < S < (3 / R, Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a + C k v k L ∞ ( B R ) S n +2 R s . Proof. If ρ ≥ S/
8, then we immediately have Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ C (cid:18) ρS (cid:19) n + a +3 Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a . Thus we may assume ρ < S/
8. Due to Lemma 3.4, we may assume v (0) >
0. Let d := dist (0 , { v ( · ,
0) = 0 } ) >
0. Then L a v = 0 in B d . Thus, if d ≥ S , we may useLemma 3.2 to obtain Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a . Thus we may also assume d < S . Case 1. S/ ≤ d ( < S ). Case 1.1.
Suppose 0 < ρ < d ( < S ). Then using L a ( ∇ x v ) = 0 in B d again, Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:16) ρd (cid:17) n + a +3 Z B d |∇ x v − h∇ x v i d | | y | a ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a . Case 1.2.
Suppose ρ ≥ d ( ≥ S/ Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:18) ρS (cid:19) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a . Case 2. < d < S/ Case 2.1.
Suppose ρ < d/
2. Take x ∈ ∂ ( B ′ d ) such that v ( x ) = 0. Then usinginclusions B ρ ⊂ B d/ ⊂ B (3 / d ( x ) ⊂ B S/ ( x ) ⊂ B R/ ( x ), L a v = 0 in B d and the preceding Lemma 3.4, we obtain Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:18) ρd (cid:19) n + a +3 Z B d/ |∇ x v − h∇ x v i d/ | | y | a ≤ (cid:18) ρd (cid:19) n + a +3 Z B (3 / d ( x ) |∇ x v − h∇ x v i x , (3 / d | | y | a ≤ (cid:18) ρd (cid:19) n + a +3 (cid:20)(cid:18) dS (cid:19) n + a +3 Z B S/ ( x ) |∇ x v − h∇ x v i x ,S/ | s | y | a + C k v k L ∞ ( B R/ ( x )) S n +2 R s (cid:21) ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a + C k v k L ∞ ( B R ) S n +2 R s Case 2.2.
Suppose d/ ≤ ρ . Then we see that B ρ ⊂ B ρ ( x ) ⊂ B S/ ( x ) ⊂ B S . Aswe did in Case 2.1, we have Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ Z B ρ ( x ) |∇ x v − h∇ x v i x , ρ | | y | a ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S/ ( x ) |∇ x v − h∇ x v i x ,S/ | | y | a + C k v k L ∞ ( B R/ ( x )) S n +2 R s ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | a + C k v k L ∞ ( B R ) S n +2 R s . (cid:3) Corollary 3.6.
Let v be a solution of the Signorini problem for L a in B R , even in y . Then there are C = C n,a , C = C n,a such that for all < ρ < S < (3 / R, Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ C (cid:16) ρS (cid:17) n + a +3 Z B S |∇ x v − h∇ x v i S | | y | + C h v i R S n +2 R s . Proof.
Since v ± = max( ± v, ≥ L a ( v ± ) = 0 in { v ± > } , we have L a ( v ± ) ≥ B R . (For this, one may follow the argument in Exercise 2.6 or Exercise 9.5 in[PSU12].) Thus, we have by Theorem 2.3.1 in [FKS82]sup B R/ v ± ≤ C (cid:18) ω n +1+ a R n +1+ a Z B R (cid:0) v ± (cid:1) | y | a (cid:19) / . Hence, k v k L ∞ ( B R/ ) ≤ C h v i R , which completes the proof. (cid:3) LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 13 Almost s -fractional harmonic functions In this section we prove Theorem I, by deducing growth estimates for almostminimizers from that of minimizers and then applying the Morrey-Campanato spaceembedding to deduce the regularity of almost minimizers.
Theorem 4.1 (Almost Lipschitz regularity) . If u is an almost s -fractional har-monic function in B ′ , < s < , then u ∈ C ,σ ( B ′ ) for any < σ < . Besides the growth estimates for minimizers we will also need the followinglemma.
Lemma 4.2.
Let r > be a positive number and let ϕ : (0 , r ) → (0 , ∞ ) be anondecreasing function. Let a , β , and γ be such that a > , γ > β > . There existtwo positive numbers ε = ε a,γ,β , c = c a,γ,β such that, if ϕ ( ρ ) ≤ a h(cid:16) ρr (cid:17) γ + ε i ϕ ( r ) + b r β for all ρ , r with < ρ ≤ r < r , where b ≥ , then one also has, still for < ρ See Lemma 3.4 in [HL97]. (cid:3) Proof of Theorem 4.1. Let K be a compact subset of B ′ containing 0. Take δ = δ n,ω,σ,K > δ < dist( K, ∂B ′ ) and ω ( δ ) ≤ ε , where ε = ε ,n +1+ a,n − a +2 σ is as Lemma 4.2. For 0 < R < δ , let v be a minimizer of Z B R |∇ v | | y | a on u + W , ( B R ). Then L a v = 0 in B R . In particular, Z B R | y | a ∇ v · ∇ ( u − v ) = 0 , and hence Z B R |∇ ( u − v ) | | y | a = Z B R |∇ u | | y | a − Z B R |∇ v | | y | a − Z B R | y | a ∇ v · ∇ ( u − v ) ≤ ω ( R ) Z B R |∇ v | | y | a . Moreover, by Lemma 3.1, for 0 < ρ < R we have Z B ρ |∇ v | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ v | | y | a . Thus Z B ρ |∇ u | | y | a ≤ Z B ρ |∇ v | | y | a + 2 Z B ρ |∇ ( u − v ) | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ v | | y | a + 2 Z B ρ |∇ ( u − v ) | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ v | | y | a + 2 ω ( R ) Z B R |∇ v | | y | a ≤ (cid:20)(cid:16) ρR (cid:17) n +1+ a + ε (cid:21) Z B R |∇ u | | y | a . By Lemma 4.2, Z B ρ |∇ u | | y | a ≤ C n,a,σ (cid:16) ρR (cid:17) n − a +2 σ Z B R |∇ u | | y | a , for any 0 < σ < 1. Taking R ր δ we have(4.1) Z B ρ |∇ u | | y | a ≤ C n,a,σ,δ k∇ u k L ( B , | y | a ) ρ n − a +2 σ . By weighted Poincar´e inequality [FKS82, Theorem (1.5)] Z B ρ | u − h u i ρ | | y | a ≤ C n,a,σ,δ k∇ u k L ( B , | y | a ) ρ n +1+ a +2 σ . Now, a similar estimates holds at all point x ∈ K , which implies the H¨oldercontinuity of u (see Theorem A.1) with k u k C ,σ ( K ) ≤ C n,a,ω,σ,K k u k W , ( B , | y | a ) . (cid:3) Theorem 4.3 ( C ,β regularity) . If u is an almost s -fractional harmonic functionin B ′ , < s < , with gauge function ω ( r ) = r α , α > , then ∇ x u ∈ C ,β ( B ′ ) forsome β = β ( n, s, α ) .Proof. Let K ⋐ B ′ be a ball and take 0 < δ < dist( K, ∂B ′ ). Let B ′ R ( x ) ⋐ B ′ with0 < R < δ , for x ∈ K . For simplicity write x = 0, and let v be the L a -harmonicfunction in B R with v = u on ∂B R . Then, by Jensen’s inequality we have Z B ρ |h∇ x u i ρ − h∇ x v i ρ | | y | a ≤ Z B ρ |∇ x u − ∇ x v | | y | a , and hence Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ Z B ρ |∇ x v − h∇ x v i ρ | | y | a + 3 Z B ρ |∇ x u − ∇ x v | | y | a + 3 Z B ρ |h∇ x u i ρ − h∇ x v i ρ | | y | a ≤ Z B ρ |∇ x v − h∇ x v i ρ | | y | a + 6 Z B ρ |∇ x u − ∇ x v | | y | a . Similarly, Z B R |∇ x v − h∇ x v i R | | y | a ≤ Z B R |∇ x u − h∇ x u i R | | y | a + 6 Z B R |∇ x u − ∇ x v | | y | a . Next let β ∈ (0 , α/ σ = 1 + β − α , we have Z B R |∇ u − ∇ v | | y | a = Z B R |∇ u | | y | a − Z B R |∇ v | | y | a ≤ R α Z B R |∇ u | | y | a ≤ C k∇ u k L ( B , | y | a ) R n +1+ a +2 β . LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 15 Then, with the help of Lemma 3.2, we have that for ρ < R Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ C Z B ρ |∇ x v − h∇ x v i ρ | | y | a + C Z B ρ |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − h∇ x v i R | | y | a + C Z B ρ |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x u − h∇ x u i R | | y | a + C Z B R |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x u − h∇ x u i R | | y | a + C k∇ u k L ( B , | y | a ) R n +1+ a +2 β . Hence, by Lemma 4.2, we obtain that for ρ < R Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ C (cid:20)(cid:16) ρR (cid:17) n +1+ a +2 β Z B R |∇ x u − h∇ x u i R | | y | a + k∇ u k L ( B , | y | a ) ρ n +1+ a +2 β (cid:21) . Taking R ր δ , we have Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ C n,a,α,β,K k∇ u k L ( B , | y | a ) ρ n +1+ a +2 β . Now, a similar estimate holds for any x ∈ K . Fixing β and applying Theorem A.1,we have k∇ x u k C ,β ( K ) ≤ C n,a,α,K k u k W , ( B , | y | a ) . (cid:3) Remark . From the assumption for almost minimizers that the Caffarelli-Silvestreextension u ∈ W , we know only that ∇ x u ∈ L , which is not sufficient to deducethe existence of the trace of ∇ x u on B ′ . However, in the proof of Theorem 4.3 weshowed that ∇ x u is in a Morrey-Campanato space, which implies the existence ofthe trace as the limit of averages T ( ∇ x u )( x ) = lim r → h∇ x u i x ,r . It is not hard to see that T ( ∇ x u ) is the distributional derivative ∇ x u on B ′ . Indeed,if η ∈ C ∞ ( B ′ ), then extending it to R n +1 by η ( x, y ) = η ( x ), we have Z B ′ T ( ∂ x i u ) η = lim r → Z B ′ h ∂ x i u i x,r η = lim r → Z B ′ ∂ x i u h η i x,r = lim r → − Z B ′ u h ∂ x i η i x,r = − Z B ′ u∂ x i η. Theorem 4.5. Let u be an almost s -fractional harmonic function in B ′ for 1, we have Z B ρ || y | a u y | | y | − a ≤ Z B ρ | v y | | y | a + 2 Z B ρ | u y − v y | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R | v y | | y | a + 2 Z B ρ | u y − v y | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R | u y | | y | a + 6 Z B R | u y − v y | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R || y | a u y | | y | − a + 6 ω ( R ) Z B R |∇ u | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R || y | a u y | | y | − a + C n,a,σ,δ ω ( R ) k∇ u k L ( B , | y | a ) R n − a +2 σ , where we used (4.1) in the last inequality.Consider now the two cases in statement of the theorem. Case 1 . 0 < s < / a > Z B ρ || y | a u y | | y | − a ≤ C (cid:20)(cid:16) ρR (cid:17) n − a +2 σ Z B R || y | a u y | | y | − a + ω ( δ ) k∇ u k L ( B , | y | a ) ρ n − a +2 σ (cid:21) ≤ C k∇ u k L ( B , | y | a ) ρ n +1 − a +( − a +2 σ ) . Now we take σ = 1 − a/ ∈ (0 , 1) to have − a + 2 σ = a > 0. Varying the center,we have a similar bound at every x ∈ K . Then, by Theorem A.1, we obtain thatthe limit of the averages T ( y | y | a − u y ) = 0 on B ′ . This implies that ( − ∆ x ) s u = 0on B ′ . Indeed, arguing as in Remark 4.4, by considering the mollifications u ε in x -variable, we note that Z B ρ || y | a ( u ε ) y | | y | − a ≤ Cρ n +1 − a + a which implies that T ( y | y | a − ( u ε ) y ) = 0 on K ⋐ B ′ . On the other hand, u ε ∈ C ∩ L s ( R n ), which implies that y | y | a − ( u ε ) y is continuous up to y = 0, since wecan explicitly write, for y > 0, the symmetrized formula y a ( u ε ) y ( x, y ) = Z R n u ε ( x + z ) + u ε ( x − z ) − u ε ( x ) | z | | z | y a ∂ y P ( z, y ) dz with locally integrable kernel | z | | y a ∂ y P ( z, y ) | ≤ C/ | z | n − − a . Hence, we obtain that( − ∆ x ) s u ε = ∂ ay u ε = 0 on the ball K ⋐ B ′ . Then, passing to the limit as ε → − ∆ x ) s u = 0 in B ′ . Case 2. s = 1 / a = 0) and ω ( r ) = r α . In this case, we have a bound Z B ρ | u y | ≤ (cid:16) ρR (cid:17) n +3 Z B R | u y | + C k∇ u k L ( B ) R n − σ + α , LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 17 Them, by Lemma 4.2, we have Z B ρ | u y | ≤ C (cid:20)(cid:16) ρR (cid:17) n − σ + α Z B R | u y | + k∇ u k L ( B ) ρ n − σ + α (cid:21) ≤ C k∇ u k L ( B ) ρ n +1+( α − σ ) . Taking 1 − α/ < σ < 1, we can guarantee that α − σ > α/ > 0, which impliesthat T ( y | y | − u y ) = 0 on B ′ . Then, arguing as at the end of Case 1, we concludethat ( − ∆ x ) / u = 0 in B ′ . (cid:3) We finish this section with formal proof of Theorem I. Proof of Theorem I. Parts (1), (2), and (3) are proved in Theorems 4.1, 4.3, and4.5, respectively. (cid:3) Almost minimizers for s -fractional obstacle problem In this section we investigate the regularity of almost minimizers for the s -fractional obstacle problem with zero obstacle and give a proof of Theorem II. Allresults in this section are proved under the assumption 1 / ≤ s < 1, or − < a ≤ Theorem 5.1 (Almost Lipschitz regularity) . Let u be an almost minimizer for s -fractional obstacle problem with zero obstacle in B ′ , for / ≤ s < . Then u ∈ C ,σ ( B ′ ) for any < σ < with k u k C ,σ ( K ) ≤ C n,a,ω,σ,K k u k W , ( B , | y | a ) for any K ⋐ B ′ .Proof. Let K ⋐ B ′ with 0 ∈ K . Take δ = δ n,a,ω,σ,K > δ < dist( K, ∂B ′ )and ω ( δ ) ≤ ε , where ε = ε ,n +1+ a,n − a +2 σ as in Lemma 4.2. For 0 < R < δ , let v be the minimizer of Z B R |∇ v | | y | a on K ,u ( B R , | y | a ). Then v satisfies the variational inequality Z B R ∇ v ∇ ( w − v ) | y | a ≥ w ∈ K ,u ( B R , | y | a ). Particularly, taking w = u , we have Z B R ∇ v ∇ ( u − v ) | y | a ≥ . As a consequence, Z B R |∇ ( u − v ) | | y | a = Z B R |∇ u | | y | a − Z B R |∇ v | | y | a + 2 Z B R | y | a ∇ v · ∇ ( v − u ) ≤ ω ( R ) Z B R |∇ v | | y | a . Next, we use (3.2) to derive a similar estimate for u . We have, Z B ρ |∇ u | | y | a ≤ Z B ρ |∇ v | | y | a + 2 Z B ρ |∇ ( u − v ) | | y | a ≤ (cid:16) ρR (cid:17) n +1+ a Z B R |∇ v | | y | a + 2 ω ( R ) Z B R |∇ v | | y | a ≤ (cid:20)(cid:16) ρR (cid:17) n +1+ a + ε (cid:21) Z B R |∇ u | | y | a . Hence, by Lemma 4.2, Z B ρ |∇ u | | y | a ≤ C n,a,σ (cid:16) ρR (cid:17) n − a +2 σ Z B R |∇ u | | y | a . As we have seen in Theorem 4.1, this implies Z B ρ |∇ u | | y | a ≤ C n,a,σ,δ k∇ u k L ( B , | y | a ) ρ n − a +2 σ (5.1)then Z B ρ | u − h u i ρ | | y | a ≤ C n,a,σ,δ k∇ u k L ( B , | y | a ) ρ n +1+ a +2 σ and ultimately k u k C ,σ ( K ) ≤ C n,a,ω,σ,K k u k W , ( B , | y | a ) . (cid:3) Theorem 5.2 ( C ,β regularity) . Let u be an almost minimizer for the s -fractionalobstacle problem with zero obstacle in B ′ , / ≤ s < , and a gauge function ω ( r ) = r α . Then ∇ x u ∈ C ,β ( B ′ ) for β < αs n +1+ a + α/ and for any K ⋐ B ′ thereholds k∇ x u k C ,β ( K ) ≤ C n,a,α,β,K k u k W , ( B , | y | a ) . Proof. Let K be a thin ball centered at 0 such that K ⋐ B . Let ε := α n +1+ a + α/ and γ := 1 − sε − ε ) . We fix R = R ( n, a, α, K ) > R − ε ≤ d/ d := dist( K, ∂B ′ ) and R < (cid:0) (cid:1) /ε . Then e K := { x ∈ B ′ : dist( x, K ) ≤ R − ε } ⋐ B . We claim that for x ∈ K and 0 < ρ < R < R ,(5.2) Z B ρ ( x ) |∇ x u − h∇ x u i x ,ρ | | y | a ≤ C n,a (cid:16) ρR (cid:17) n + a +3 Z B R ( x ) |∇ x u − h∇ x u i x ,R | | y | a + C n,a,α,K k u k W , ( B , | y | a ) R n +1+ a + sε . Note that once we have this bound, the proof will follow by the application ofLemma 4.2 and Theorem A.1.For simplicity we may assume x = 0, and fix 0 < R < R . Let R := R − ε . Let v be the minimizer of Z B R |∇ v | | y | a on K ,u ( B R , | y | a ). Then by (3.2) and (5.1) with σ = γ , for 0 < ρ ≤ R (5.3) Z B ρ |∇ v | | y | a ≤ (cid:18) ρR (cid:19) n +1+ a Z B R |∇ v | | y | a ≤ (cid:18) ρR (cid:19) n +1+ a Z B R |∇ u | | y | a ≤ C n,a,α,K (cid:18) ρR (cid:19) n +1+ a k u k W , ( B , | y | a ) R n − a +2 γ ≤ C n,a,α,K k u k W , ( B , | y | a ) ρ n − a +2 γ . LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 19 This gives(5.4) − Z B ρ | v − v ρ | | y | a ≤ C k u k W , ( B , | y | a ) ρ γ , C = C n,a,α,K . Since this estimate holds for any 0 < ρ < R , the standard dyadic argument gives(5.5) | v (0) − h v i R | ≤ C k u k W , ( B , | y | a ) R γ , C = C n,a,α,K . Moreover, using (3.2) and (5.1) again, we have for any x ∈ B ′ R/ , 0 < ρ < R/ Z B ρ ( x ) |∇ v | | y | a ≤ (cid:18) ρR (cid:19) n +1+ a Z B R/ ( x ) |∇ v | | y | a ≤ (cid:18) ρR (cid:19) n +1+ a Z B R |∇ u | | y | a ≤ C n,a,α,K k u k W , ( B , | y | a ) ρ n − a +2 γ , which implies(5.7) [ v ] C ,γ ( B ′ R/ ) ≤ C k u k W , ( B , | y | a ) , C = C n,a,α,K . Now we define C := C + C + C . Our analysis then distinguishes the following two cases h v i R ≤ C k u k W , ( B , | y | a ) R γ or h v i R > C k u k W , ( B , | y | a ) R γ . Case 1. Suppose first that h v i R ≤ C k u k W , ( B , | y | a ) R γ . Note that R < (cid:0) (cid:1) /ε implies R < R . Then, using Corollary 3.6, we see thatfor 0 < ρ < R , Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ Z B ρ |∇ x v − h∇ x v i ρ | | y | a + 6 Z B ρ |∇ x u − ∇ x v | | y | a dx ≤ C n,a (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − h∇ x v i R | | y | a + C n,a h v i R R n +2 R s + 6 Z B ρ |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x u − h∇ x u i R | | y | a + C h v i R R n +2 R s + C Z B R |∇ x u − ∇ x v | | y | a . Note that for σ := 1 − α/ Z B R |∇ x u − ∇ x v | | y | a ≤ Z B R |∇ x u − ∇ x v | | y | a ≤ R α Z B R |∇ v | | y | a ≤ R α Z B R |∇ u | | y | a ≤ C n,a,α,K R α k u k W , ( B , | y | a ) R n − a +2 σ = C k u k W , ( B , | y | a ) R n +1+ a + α/ . Moreover by the assumption C h v i R R n +2 R s ≤ C n,a,α,K k u k W , ( B , | y | a ) R n +2 R γ − − s = C k u k W , ( B , | y | a ) R n +1+ a + sε . Hence, we obtain (5.2) in this case. Case 2 . Now we assume h v i R > C k u k W , ( B , | y | a ) R γ . Then, by (5.4) and (5.5) we obtain − Z B R | v − v (0) | | y | a ≤ − Z B R | v − v R | | y | a + 2 − Z B R | v R − v (0) | | y | a ≤ C k u k W , ( B , | y | a ) R γ . Combining the latter bound and the assumption, v (0) = − Z B R | v (0) | | y | a ≥ / − Z B R | v ( X ) | | y | a − − Z B R | v ( X ) − v (0) | | y | a ≥ C k u k W , ( B , | y | a ) R γ . Since C ≥ C , we have v > B ′ R/ by (5.7). Thus, L a v = 0 in B R/ , and byLemma 3.2 we have for 0 < ρ < R Z B ρ |∇ x v − h∇ x v i ρ | | y | a ≤ (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − h∇ x v i R | | y | a . Thus, Z B ρ |∇ x u − h∇ x u i ρ | | y | a ≤ Z B ρ |∇ x v − h∇ x v i ρ | | y | a + 6 Z B ρ |∇ x u − ∇ x v | | y | a ≤ (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x v − h∇ x v i R | | y | a + 6 Z B ρ |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x u − h∇ x u i R | | y | a + C Z B R |∇ x u − ∇ x v | | y | a ≤ C (cid:16) ρR (cid:17) n + a +3 Z B R |∇ x u − h∇ x u i R | | y | a + C k u k W , ( B , | y | a ) R n +1+ a + α/ . This implies (5.2) and completes the proof. (cid:3) LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 21 Proof of Theorem II. Parts (1) and (2) are contained in Theorems 5.1 and 5.2,respectively. (cid:3) Appendix A. Morrey-Campanato-type Space Theorem A.1. Let u ∈ L ( B , | y | a ) and M be such that k u k L ( B , | y | a ) ≤ M andfor some σ ∈ (0 , Z B r ( x ) | u − h u i x,r | | y | a ≤ M r n +1+ a +2 σ , h u i x,r = 1 ω n +1+ a r n +1+ a Z B r ( x ) u | y | a for any ball B r ( x ) centered at x = ( x, ∈ B ′ / and radius < r < r ≤ / . Thenfor any x ∈ B ′ / there exists the limit of averages T u ( x ) := lim r → h u i x,r , which will also satisfy Z B r ( x ) | u − T u ( x ) | | y | a ≤ C n,a,σ M r n +1+ a +2 σ . Moreover, T u ∈ C ,σ ( B ′ / ) with k T u k C ,σ ( B ′ / ) ≤ C n,a,σ,r M. Remark A.2 . Note, we can redefine u ( x, 0) = T u ( x ) for any x ∈ B ′ / , making ( x, u . Proof. Let x, z ∈ B ′ / and 0 < ρ < r < r be such that B ρ ( x ) ⊂ B r ( z ). Then |h u i x,ρ − h u i z,r | ≤ − Z B ρ ( x ) | u − h u i z,r || y | a ≤ (cid:18) rρ (cid:19) n +1+ a − Z B r ( z ) | u − h u i z,r || y | a ≤ (cid:18) rρ (cid:19) n +1+ a − Z B r ( z ) | u − h u i z,r | | y | a ! / − Z B r ( z ) | y | a ! / ≤ C n,a (cid:18) rρ (cid:19) n +1+ a M r σ . Now, taking x = z and using a dyadic argument, we can conclude that |h u i x,ρ − h u i x,r | ≤ C n,a,σ M r σ , for any 0 < s = ρ < r < r . Indeed, let k = 0 , , , . . . be such that r/ k +1 ≤ ρ < r/ k . Then |h u i x,ρ − h u i x,r | ≤ k X j =1 |h u i x,r/ j − − h u i x,r/ j | + |h u i x,r/ k − h u i x,ρ |≤ C n,a M k +1 X j =1 ( r/ j − ) σ ≤ C n,a,σ M r σ . This implies that the limit T u ( x ) = lim r → h u i x,r exists and | T u ( x ) − h u i x,r | ≤ C n,a,σ M r σ . Hence, we also have the H¨older integral bound Z B r ( x ) | u − T u ( x ) | | y | a ≤ C n,a,σ M r n +1+ a +2 σ . Besides, we have | T u ( x ) | ≤ h u i x,r + C n,a,σ M r σ ≤ C n,a,σ,r M. It remains to estimate the H¨older seminorm of T u on B ′ / . Let x, z ∈ B ′ / andconsider two cases. Case 1. If | x − z | < r / 4, let r = 2 | x − z | . Then note that B r/ ( x ) ⊂ B r ( z ) andtherefore we can write | T u ( x ) − T u ( z ) | ≤ | T u ( x ) − h u i x,r/ | + | T u ( z ) − h u i z,r | + |h u i x,r/ − h u i z,r |≤ C n,a,σ M r σ = C n,a,σ M | x − z | σ . Case 2. If | x − z | ≥ r / 4, then | T u ( x ) − T u ( z ) | ≤ | T u ( x ) | + | T u ( z ) |≤ C n,a,σ,r M ≤ C n,a,σ,r M | x − z | σ . Thus, we conclude k T u k C ,σ ( B ′ / ) ≤ C n,a,σ,r M. (cid:3) Appendix B. Polynomial expansion for Caffarelli-Silvestre extension Some of the results in Section 3 rely on polynomial expansion theorem for L a -harmonic functions given below. Theorem B.1. Let u ∈ W , ( B , | y | a ) , − < a < , be a weak solution of the equa-tion L a u = 0 in B , even in y . Then we have the following polynomial expansion: u ( x, y ) = ∞ X k =0 p k ( x, y ) locally uniformly in B , where p k ( x, y ) are L a -harmonic polynomials, homogeneousof degree k and even in y . Moreover, the polynomials p k above are orthogonal in L ( ∂B , | y | a ) , i.e., Z ∂B p k p m | y | a = 0 , k = m. In, particular, u is real analytic in B . This theorem has the following immediate corollaries, which are of independentinterest and are likely known in the literature. We state them here for reader’sconvenience and for possible future reference. Corollary B.2. Let u ∈ W , ( B , | y | a ) , − < a < , be a weak solution of theequation L a u = 0 in B . Then, we have a representation u ( x, y ) = ϕ ( x, y ) + y | y | − a ψ ( x, y ) , ( x, y ) ∈ B , where ϕ ( x, y ) and ψ ( x, y ) are real analytic functions, even in y . LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 23 Corollary B.3. Let u ∈ L s ( R n ) satisfies ( − ∆) s u = 0 in the unit ball B ′ ⊂ R n .Then u is real analytic in B ′ . Corollary B.4. Let u ∈ W , ( B , | y | a ) , − < a < , be a weak solution of theequation L a u = 0 in B , even in y . If u ( · , ≡ in B ′ , then u ≡ in B . The proof of Theorem B.1 and subsequently those of Corollaries B.2, B.3, andB.4 are based on the following lemmas. We follow the approach of [ABR01] forharmonic functions.Let P ∗ m = { p : p ( x, y ) polynomial of degree ≤ m , even in y } . Lemma B.5. Let p ∈ P ∗ m . Then there exists ˜ p ∈ P ∗ m such that L a ˜ p = 0 in B , ˜ p = p on ∂B . In other words, the solution of the Dirichlet problem for L a in B with boundaryvalues in P ∗ m on ∂B is itself in P ∗ m .Proof. For m = 0 , 1, we simply have ˜ p = p . For m ≥ 2, we proceed as follows.For q ∈ P ∗ m − define T q ∈ P ∗ m − by( T q )( x, y ) = | y | − a L a ((1 − x − y ) q ( x, y )) . (It is straightforward to verify that T q is indeed in P ∗ m − ). We now claim that themapping T : P ∗ m − → P ∗ m − is bijective. Since T is clearly linear and P ∗ m − is finitedimensional it is equivalent to showing that T is injective. To this end, supposethat T q = 0 for some q ∈ P ∗ m − . This means that Q ( x, y ) = (1 − x − y ) q ( x, y ) is L a -harmonic in B : L a Q = 0 in B . On the other hand Q = 0 on ∂B and therefore, by the maximum principle Q = 0in B . But this implies that q = 0 in B , or that q ≡ 0. Hence, the mapping T isinjective, and consequently bijective. It is now easy to see that˜ p = p − (1 − x − y ) T − ( | y | − a L a ( p )) ∈ P ∗ m satisfies the required properties. (cid:3) Lemma B.6. Polynomials, even in y , are dense in the subspace of functions in L ( ∂B , | y | a ) , even in y .Proof. Polynomials, even in y are dense in the space of continuous functions in C ( ∂B ), even in y , with the uniform norm. The claim now follows from the obser-vation that the embedding C ( ∂B ) ֒ → L ( ∂B , | y | a ) is continuous: k v k L ( ∂B , | y | a ) ≤ k v k L ∞ ( ∂B ) (cid:18)Z ∂B | y | a (cid:19) / ≤ C k v k L ∞ ( ∂B ) . (cid:3) Lemma B.7. The subspace of functions in L ( ∂B , | y | a ) , even in y , has an or-thonormal basis { p k } ∞ k =0 consisting of homogeneous L a -harmonic polynomials p k ,even in y .Proof. If p is a polynomial, even in y , then restricted to ∂B it can be replacedwith an L a -harmonic polynomial ˜ p . On the other hand, if we decompose˜ p = m X i =0 q i where q i is a homogeneous polynomial of order i , even in y , then | y | − a L a ˜ p = m X i =2 | y | − a L a q i where | y | − a L a q i is a homogeneous polynomial of order i − i = 2 , . . . , m . Hence, L a ˜ p = 0 iff L a q i = 0, for all i = 0 , . . . , m (for i = 0 , q i and q j are two homogeneous L a -harmonic polynomialsof degrees i = j , then they are orthogonal in L ( ∂B , | y | a ). Indeed,0 = Z B q i div( | y | a ∇ q j ) − div( | y | a ∇ q i ) q j = Z ∂B ( q i ∂ ν q j − q j ∂ ν q i ) | y | a = ( j − i ) Z ∂B q i q j | y | a . Using this and following the standard orthogonalization process, we can constructa basis consisting of homogeneous L a -harmonic polynomials. (cid:3) Lemma B.8. Let u ∈ W , ( B , | y | a ) ∩ C ( B ) is a weak solution of L a u = 0 in B .Then k u k L ∞ ( K ) ≤ C n,a,K k u k L ( ∂B , | y | a ) . for any K ⋐ B .Proof. First, we note that by [FS87] k u k L ∞ ( K ) ≤ C n,a,K k u k L ( B , | y | a ) . So we just need to show that k u k L ( B , | y | a ) ≤ C n,a k u k L ( ∂B , | y | a ) . This follows from the fact that u is a subsolution: L a ( u ) ≥ B and thereforethe weighted spherical averages r ω n,a r n + a Z ∂B r u | y | a , < r < k u k L ( B , | y | a ) ≤ C n,a k u k L ( ∂B , | y | a ) . (cid:3) We are now ready to prove Theorem B.1. Proof of Theorem B.1. Without loss of generality we may assume u ∈ W , ( B , | y | a ) ∩ C ( B ), otherwise we can consider a slightly smaller ball. Now, using the orthonor-mal basis { p k } ∞ k =0 from Lemma B.7 we represent u = ∞ X k =0 a k p k in L ( ∂B , | y | a ) . We then claim that u ( x, y ) = ∞ X k =0 a k p k ( x, y ) uniformly on any K ⋐ B . Indeed, if u m ( x, y ) = P mk =0 a k p k ( x, y ), then k u − u m k L ( ∂B , | y | ) → m → ∞ and therefore by Lemma B.8 k u − u m k L ∞ ( K ) ≤ C n,a,K k u − u m k L ( ∂B , | y | a ) → . (cid:3) LMOST MINIMIZERS FOR CERTAIN FRACTIONAL VARIATIONAL PROBLEMS 25 We now give the proofs of the corollaries. Proof of Corollary B.2. Write u ( x, y ) in the form u ( x, y ) = u even ( x, y ) + u odd ( x, y ) , where u even and u odd are even and odd in y , respectively. Clearly, both functionsare L a -harmonic. Moreover, by Theorem B.1, u even is real analytic and we take ϕ = u even . On the other hand, consider v ( x, y ) = | y | a ∂ y u odd ( x, y ) . Then, v is L − a -harmonic in B and again by Theorem B.1, v is real analytic. Wecan now represent u odd ( x, y ) = y | y | − a ψ ( x, y ) , ψ ( x, y ) = y − | y | a Z y | s | − a v ( x, s ) ds. It is not hard to see that ψ ( x, y ) is real analytic, which completes our proof. (cid:3) Proof of Corollary B.3. The proof follows immediately from Theorem B.1 by con-sidering the Caffarelli-Silvestre extension u ( x, y ) = u ∗ P ( · , y ) = Z R n P ( x − z, y ) u ( z ) dz, ( x, y ) ∈ R n × R + where P ( x, y ) = C n,a y − a ( | x | + y ) ( n +1 − a ) / is the Poisson kernel for L a , and noting thatits extension to R n +1 by even symmetry in y (still denoted u ) satisfies L a u = 0 in B . (cid:3) Proof of Corollary B.4. Represent u ( x, y ) as a locally uniformly convergent in B series u ( x, y ) = ∞ X k =0 q k ( x, y ) , where q k ( x, y ) is a homogeneous of degree k L a -harmonic polynomial, even in y .We have u ( x, 0) = ∞ X k =0 q k ( x, ≡ q k ( x, ≡ 0. We now want to show that q k ≡ 0. Tothis end represent q k ( x ) = [ k/ X j =0 p k − j ( x ) y j , where p k − j ( x ) is a homogeneous polynomial of order k − j in x . Clearly p k ( x ) ≡ ∂ αx q k ( x ) of order | α | = k − 2, we see that ∂ αx q k ( x ) = c α y , c α = ∂ αx p k − is L a -harmonic, which can happen only when c α = 0. 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MR2962060[Sil07] Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplaceoperator , Comm. Pure Appl. Math. (2007), no. 1, 67–112, doi:10.1002/cpa.20153.MR2270163 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail address , S.J.: [email protected] Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail address , A.P.:, A.P.: . Then u isactually s -fractional harmonic in B ′ .Proof. We argue as in the proof Theorem 4.1. Let K , δ , R , v be as in the proof ofthat theorem. Then, by Lemma 3.1, for 0 < ρ < R Z B ρ | v y | | y | a ≤ (cid:16) ρR (cid:17) n +3+ a Z B R | v y | | y | a . Thus, for any 0 < σ <