Minimum principles and a priori estimates for some translating soliton type problems
aa r X i v : . [ m a t h . DG ] D ec Minimum principles and a priori estimates forsome translating soliton type problems
Rafael L´opezDepartamento de Geometr´ıa y Topolog´ıaInstituto de Matem´aticas (IEMath-GR)Universidad de Granada18071 Granada, Spain [email protected]
Cristian EnacheAmerican University of SharjahDepartment of Mathematics and StatisticsUniversity City Road, P.O. Box 26666, Sharjah, UAE [email protected]
Abstract
In this paper we are dealing with two classes of mean curvature type prob-lems that generalize the translating soliton problem. A first result proves thatthe solutions to these problems have unique interior critical points. Using thisuniqueness result, we next derive a priori C and C estimates for the solu-tions to these problems, by means of some minimum principles for appropriate P -functions, in the sense of L.E. Payne. Keywords: critical points, nodal lines, maximum principles, a priori estimates
MSC 2010:
This paper is devoted to the study of the following two general classes of meancurvature type problems: 1 div ∇ u p |∇ u | ! = p |∇ u | ! α in Ω ,u = 0 on ∂ Ω , (1)(2) div ∇ v p |∇ v | ! = 1 p |∇ v | + µ in Ω ,v = 0 on ∂ Ω , (3)(4)where Ω ⊂ R is a bounded strictly convex domain with smooth boundary ∂ Ω, while α, µ > κ = κ ( s ) denotes the curvatureof ∂ Ω as a planar curve in R computed with respect to the inward orientation, thenthe strictly convexity of Ω is equivalent to κ > ∂ Ω.The motivation for considering these problems has its origin in the singularity theoryof the mean curvature flow in R of Huisken and Ilmanen [8, 9, 32]. A translatingsoliton is a surface Σ ⊂ R that is a solution of the mean curvature flow when Σevolves purely by translations along some direction ~a ∈ R \ { } . For the initialsurface Σ in the flow, this implies that 2 H = h N, ~a i , where N is a choice of unitnormal field. After a change of coordinates, we suppose ~a = (0 , , z = u ( x, y ), then the identity 2 H = h N, ~a i is now rewritten asdiv ∇ u p |∇ u | ! = 1 p |∇ u | . (5)Equation (5) is called the translating soliton equation (see Lopez [12] for a historicalintroduction of this equation). Therefore, equations (1) and (3) generalize (5), bytaking α = 1 in (1) and µ = 0 in (3), respectively. However, both equations (1)and (3) have their own interest. Equation (1) has been considered in [25, 26, 29]as the extension of the flow of surfaces by powers of the mean curvature H . Onthe other hand, equation (3) is better understood when we see a solution of (5) inthe context of manifolds with density ([7, 16]). More precisely, let e ϕ be a positivedensity function in R , with ϕ ∈ C ∞ ( R ), which serves as a weight for the volumeand the surface area. For a given variation Σ t of Σ, let us denote by A ϕ ( t ) and V ϕ ( t )the weighted area and the enclosed weighted volume of Σ t , respectively. Then theexpressions of the first variation of A ϕ ( t ) and V ϕ ( t ) are A ′ ϕ (0) = − Z Σ H ϕ h N, ξ i dA ϕ , V ′ ϕ (0) = Z Σ h N, ξ i dA ϕ , (6)where ξ is the variational vector field of Σ t and H ϕ = H − h∇ φ, N i / ϕ ( q ) = h q, ~a i , where ~a = (0 , , ϕ = H − h N, ~a i /
2. As a consequence of the Lagrange multipliers, we conclude thatΣ is a critical point of the area A ϕ for a prescribed weighted volume if and only if H ϕ is identically constant, H ϕ = µ/
2: this equation coincides with (3) when Σ isthe graph of z = u ( x, y ). For µ = 0, the solutions of equation (3) invariant by auniparametric family of rigid motions have been classified by the second author in[11]. In a general context, there is a great interest of the solvability of the meancurvature equation (3) by replacing the constant µ by a ‘forcing term’ f = f ( u, Du )(see [2, 10, 15, 24]).In this paper we will not study the existence of solutions to problems (1)-(2) and(3)-(4). Sufficient conditions on the data for the existence of classical solutions areknown in the above bibliography, or more generally, in the classical article of Ser-rin [27]. Here we are rather interested in obtaining estimates for the solutions toboth Dirichlet problems. To this end, we will extensively use the theory of maxi-mum principles developed by L.E. Payne and G.A Philippin in [19] for quasilinearelliptic equations of divergence type (see also the book of R. Sperb [30] and thereferences therein). We will thus develop some new minimum principles for two so-called P -functions in the sense of L.E. Payne, that is, for two appropriate functionalcombinations of the solutions and their derivatives. More precisely, let us considerthe following P -functions:Φ( x ; β ) = 2 α − (cid:0) |∇ u | (cid:1) α − − βu, (7)Ψ( x ; β ) = ln |∇ v | (cid:16) µ p |∇ v | (cid:17) − βv, (8)with β ∈ R , where u and v are solutions to problems (1)-(2) and (3)-(4), respectively.Moreover, let us also assume that α = 1, since the case α = 1, namely, the translatingsoliton equation (5), was already investigated by Barbu and Enache [1]. The mainresult of this paper is the following minimum principle. Theorem 1.1. If β ∈ [1 , , then the auxiliary functions Φ( x ; β ) and Ψ( x ; β ) attaintheir minimum values on the boundary ∂ Ω . As a consequence of these minimum principles for Φ( x ; β ) and Ψ( x ; β ), we derivethe following a priori estimates. Theorem 1.2. If u is a solution of problem (1)-(2), with α = 1 , then we have thefollowing lower bound estimates: q min ≥ ( κ max ) − α +1 , (9)3 u min ≥ α − (cid:18) κ max (cid:19) α − α +1 − ! , (10) where q min = min ∂ Ω |∇ u | , u min = min Ω u and κ max = max ∂ Ω κ . Theorem 1.3. If v is a solution of problem (3)-(4), then we have the followinglower bound estimates: q min ≥ µ κ max , (11) − v min ≥ (1 + µ ) q (1+ µ ) κ µ q (1+ µ ) κ , (12) where q min = min ∂ Ω |∇ v | , v min = min Ω v and κ max = max ∂ Ω κ . Minimum principles for appropriate P -functions, similar to our results, have beenobtained for several problems of physical or geometrical interests: [4, 5, 13, 14, 18,20, 21, 22]. While the corresponding maximum principles are usually easier to obtainas, for instance, in the reference paper [19], the minimum principles usually requiresome additional properties of solutions, such as the convexity of the level curves ofthe solutions or the the uniqueness of their critical points. For the two problems ofthis paper, the convexity of level sets of the solutions is still unknown. However, weare able to show that the solutions have an unique critical point, which allows us toadapt a technique employed in [20] for the case α = 0 in (1), and in [1] for the case µ = 0 in (3).The paper is organized as follows. In Sections 2 and 3 we will give the proof ofTheorem 1.1 for problems (1)-(2) and (3)-(4), respectively, together with some pre-liminary results. In Section 4 we will apply Theorems 1.1 to derive the lower boundestimates from Theorem 1.2, while in Section 5 we will apply again Theorem 1.1,to obtain the lower bound estimates from Theorem 1.3. Finally, in Sections 6, weshow that some maximum principles developed by Payne and Philippin in [19] canbe also employed to obtain upper bound estimates which complement the resultsfrom Theorems 1.2 and 1.3 (see Theorems 6.1 and 6.2). For the proof of Theorem 1.1, we need first to investigate the number of criticalpoints of the solution to problem (1)-(2). We point out that the study of the numbercritical points of solutions for elliptic problems is a subject of high interest and the4iterature is very extensive: here we only refer [23] in the context of the constantmean curvature equation.Since the right hand-side of (1) is positive, the strong maximum principle impliesthat u < u attains its minimum at some interior point of Ω. The nextresult has its own interest and proves that, in fact, there is only one interior criticalpoint for the solution to problem (1)-(2). Theorem 2.1.
The solution u of problem (1)-(2) has only one critical point in Ω .Proof. The proof follows the arguments used by Philippin in [20], for the case α = 0in problem (1)-(2). For completeness, we also give it briefly here.Before starting the proof, let us note that, in what follows in this paper, we willalways employ the summation convention over repeated indices (from 1 to 2) andadopt the following notations: u = ∂u∂x , u = ∂u∂x , u ij = ∂ u∂x i ∂x j , for i, j ∈ { , } . As for the proof, a first observation is that the solution u of (1)-(2) is analytic inΩ (see Nirenberg [17]). We denote z k = u k , k = 1 ,
2, and write equation (1) in theform (cid:0) |∇ u | (cid:1) ∆ u − u ij u i u j = (cid:0) |∇ u | (cid:1) − α . (13)Differentiating (13) with respect to x k , k = 1 ,
2, we see that both z and z satisfythe differential equation (cid:0)(cid:0) |∇ u | (cid:1) δ ij − u i u j (cid:1) z kij + 2 (cid:18) u i ∆ u − u ij u i − − α (cid:0) |∇ u | (cid:1) − α u i (cid:19) z ki = 0 , (14)in Ω, where δ ij is the Kronecker symbol. Since equation (14) is linear in z , a linearcombination of z and z of type z ( θ ) = z cos θ + z sin θ, with θ ∈ R , also satisfies equation (14) in Ω. Therefore, the strong maximumprinciple implies that z takes its minimum and maximum values on ∂ Ω ([6, Cor.3.2]). On the other hand, since u = 0 on ∂ Ω, we have z k = ∂u∂ n n k on ∂ Ω , where n = ( n , n ) is the outward unit normal vector on ∂ Ω and ∂u/∂ n is theoutward normal derivative of u . Then z ( θ ) can be now rewritten as z ( θ ) = ∂u∂ n n · (cos θ, sin θ ) on ∂ Ω . u < ∂u∂ n < ∂ Ω . Let e iθ be a fixed arbitrary direction in the plane R . Since ∂ Ω is strictly convex,the normal map n : ∂ Ω → S is one-to-one on the unit circle S . We thus deducethat n ( s ) is orthogonal to e iθ at exactly two points and by the definition of z ( θ ),the function z ( θ ) vanishes along ∂ Ω at exactly two points.The proof of Theorem 2.1 is obtained by contradiction. Suppose that there exist atleast two critical points of u in Ω, namely, P and P . Then:1. The function z ( θ ) is not constant in Ω because z ( θ ) has only two zeros along ∂ Ω. Since z ( θ ) is analytic, the critical points of z ( θ ) are isolated points.2. Let N θ = z ( θ ) − (0) be the nodal set of z ( θ ). Since z ( θ ) is analytic, standardtheory asserts that near a critical point of z ( θ ), the function z ( θ ) is asymptoti-cally approximated by a harmonic homogeneous polynomial. Following Cheng[3], N θ is diffeomorphic to the nodal set of the approximating homogeneouspolynomial. In particular, N θ is formed by a set of regular analytic curves atregular points, the so-called nodal lines. On the other hand, in a neighborhoodof a critical point, the nodal lines form an equiangular system.We point out that there is no closed component of N θ contained in Ω. Indeed,if we assume that N θ encloses a subdomain Ω ′ of Ω, then z ( θ ) = 0 along ∂ Ω ′ , so the maximum principle would imply that z ( θ ) is identically 0 in Ω ′ ,contradicting the fact that z ( θ ) is not constant.3. We prove that N θ is formed from only one nodal line. Suppose by contradictionthat there exist two nodal lines L and L . Since L an L are not closed,then the arcs L and L end precisely at the two boundary points where z ( θ )vanishes. Since Ω is simply-connected, then L and L enclose at least onesubdomain Ω ′ ⊂ Ω: this is impossible by the previous item.4. As a conclusion, the nodal set N θ is formed from exactly one arc. We now givean orientation to the arc N θ for each θ : the orientation of N θ is chosen such thatwe first pass through P and then through P . With respect to this orientation,we are ordering the two boundary points where z ( θ ) vanishes. More precisely,let us denote by P ( θ ) the initial point of N θ , which after passing P and then P , finishes at the other boundary point, which is denoted by Q ( θ ).5. Let us consider θ varying in the interval [0 , π ]. By the definition of z ( θ ), thefunctions z (0) and z ( π ) coincides up to the sign, that is, z (0) = − z ( π ) and6hus the nodal lines N and N π coincide as sets of points. However, when θ runs in [0 , π ], the ends points of N interchange their position when θ reachesthe value θ = π , leading to the nodal line N π . Therefore, according to thechosen orientation in N θ , P (0) = Q ( π ) and P ( π ) = Q (0). Since all the arcs N θ pass first through P and then through P , this interchange of the end pointsbetween N and N π would imply the existence of another nodal line for z ( π ).But this is impossible, by item 3. This contradiction completes the proof ofTheorem 2.1.Now, once the uniqueness of the interior critical point of u is proved, using somerotation and/or translation if necessary, we can choose the coordinates axes suchthat the unique critical point of u is located at O , the origin of the coordinate system.Then O is the unique point of global minimum for u , so we have u ( O ) ≥ u ( O ) ≥
0. The next lemma shows that in fact these inequalities are strict.
Lemma 2.2. If u is the solution of problem (1) - (2) , then u ( O ) > , u ( O ) > . Proof.
The proof is obtained by contradiction. Suppose that u ( O ) = 0 (a similarargument will work if we assume instead that u ( O ) = 0). If the function z = u is constant in Ω, then u depends only on the variable x and the boundary condition(2) is impossible. Thus z is a non constant analytic function. Since z vanishes at O as well as z and z , then the function z vanishes at O with a finite order m ≥ z which form an equiangularsystem in a neighborhood of O . However we have already proved in Theorem 2.1the existence of exactly one nodal line, unless z is constant in Ω, so that we achievea contradiction. Lemma 2.3. If β ∈ [1 , , then the auxiliary function Φ( x ; β ) attains its minimumvalue at the critical point of u or on the boundary ∂ Ω .Proof. Differentiating successively (7), we haveΦ k = 2 (cid:0) |∇ u | (cid:1) α − u ik u i − βu k , (15)respectivelyΦ kl = 2 ( α − (cid:0) |∇ u | (cid:1) α − u ik u i u jl u j + 2 (cid:0) |∇ u | (cid:1) α − ( u ikl u i + u il u il ) − βu kl . (16)7e now remind the following identity u ik u ik |∇ u | = |∇ u | (∆ u ) + 2 u ij u i u kj u k − u ) u ij u i u j , (17)which holds only in R (see [22]). Making use of (17), after some manipulations (see[19, Eq. (2.15)]), we obtain∆Φ −
11 + |∇ u | Φ kj u k u j + W k Φ k = ( β − (cid:0) |∇ u | (cid:1) − α +12 (cid:16) β − β α |∇ u | (cid:17) , (18)where W k is a smooth vector function which is singular at the critical point of u . Weobserve that the right hand-side of (18) is non-positive, because β − ≤
0, and theother two parentheses are positive. Therefore, the conclusion of Lemma 2.3 followsnow from (18), as a direct consequence of the strong maximum principle.
Lemma 2.4. If β ∈ [1 , , then the auxiliary function Φ( x ; β ) cannot be identicallyconstant on Ω .Proof. If β ∈ [1 , x ; β ) can satisfy (18) because theright hand-side is positive. Therefore, it remains to investigate the case β = 2. Insuch a case, we assume contrariwise that Φ( x ; 2) is constant on Ω. By the definitionof Φ( x ; 2) and the fact that u = 0 on ∂ Ω, we deduce that |∇ u | is constant on ∂ Ω.Therefore, according to a symmetry result of Serrin ([28]), the domain Ω must bea disk and the solution to problem (1)-(2) must be radial, that is, u = u ( r ), with r = | x | . Now, in radial coordinates, equation (1) can be rewritten as u rr + 1 r u r (1 + u r ) = (1 + u r ) − α . (19)On the other hand, since Φ( x ; 2) is constant, we have that ∂ Φ /∂r = Φ ,k u ,k = 0, sothat u rr = (1 + u r ) − α and (19) becomes1 r u r (1 + u r ) = 0 , which is impossible, since u r = 0 for r = 0. We have thus obtained a contradictionand the proof is achieved.We are now in position to prove Theorem 1.1. The proof is obtained by contradiction.Let us assume that the minimum of Φ( x ; β ) occurs at the critical point O of u . Wedistinguish two cases. 8. Case β ∈ (1 , u ( O ) = u ( O ) = u ( O ) = 0, we evaluate(15) and (16) at O to obtainΦ ( O ; β ) = Φ ( O ; β ) = 0 , respectively Φ ( O ; β ) = 2 u ( O ) − βu ( O ) , Φ ( O ; β ) = 0 , Φ ( O ; β ) = 2 u ( O ) − βu ( O ) . Since Φ( x ; β ) attains its minimum in O , we have0 ≤ Φ ( O ; β ) = u ( O )(2 u ( O ) − β ) , ≤ Φ ( O ; β ) = u ( O )(2 u ( O ) − β ) . (20)It follows then from Lemma 2.2 and (20) that2 u ( O ) − β ≥ , u ( O ) − β ≥ . (21)Summing now these two last inequalities, if follows that∆ u ( O ) − β ≥ . (22)On the other hand, evaluating (1) at O , we get∆ u ( O ) = 1 . (23)Inserting now (23) into (22), we conclude that β ≤
1, which contradicts theassumption that β >
1, so that Theorem 1.1 is proved in this case.2. Case β = 1. We repeat a continuity argument used by Philippin and Safoui in[22]. From Lemma 2.4 we know that for all β ∈ [1 ,
2] the auxiliary functionsΦ( x ; β ) take its minimum value either on the boundary ∂ Ω or at the criticalpoint of u . On the other hand, from the previous case β <
1, we also knowthat Φ( x ; β ) takes its minimum value on ∂ Ω for all β ∈ (1 , β decreases continuously from 2 to 1, the points at which Φ( x ; β ) takesits minimum value have to move continuously. Therefore, they cannot jumpaway from ∂ Ω at the interior critical point of u . This contradiction thus provesTheorem 1.1, when β = 1. The proof of Theorem 1.1 for problem (3)-(4) is similar to the one given in theprevious section for problem (1)-(2). We will thus follow the same steps and onlypresent the differences. Recall that the constant µ in (3) is positive.9gain, let us notice that, since the right hand-side of equation (3) is positive, thestrong maximum principle implies that the solution v satisfies v < v has only one critical point in Ω. To this end, the argument isthe same as the one from the previous section. More precisely, we first differentiateequation (3) with respect to x k , to obtain (cid:0)(cid:0) |∇ v | (cid:1) δ ij − v i v j (cid:1) z kij + 2 (cid:18) v i ∆ v − v ij u i − v i − µ (1 + |∇ v | ) (cid:19) z ki = 0 . Therefore, the strong maximum principle can be applied to this equation and theremaining part of the proof is identical to what one has already seen in the proofof Theorem 2.1. This means that v has a unique critical point, which is a pointof global minimum. As before, we may assume that this point is the origin O andclearly a result similar to Lemma 2.2 can be easily derived in this case, to obtainthat v ( O ) > v ( O ) > Lemma 3.1. If β ∈ [1 , , then the auxiliary function Ψ ( x ; β ) attains its minimumvalue at the critical point of v or on the boundary ∂ Ω .Proof. Differentiating successively (8), we obtainΨ k = 2 v ik v i (1 + |∇ v | ) (cid:16) µ p |∇ v | (cid:17) − βv k , (24)respectivelyΨ kl = 2 v ikl v i + 2 v ik v il |∇ v | + µ (1 + |∇ v | ) / − (cid:16) µ p |∇ v | (cid:17) v ik v i v jl v j (1 + |∇ v | + µ (1 + |∇ v | ) / ) − βv kl . (25)Then the equation corresponding here to (18) is∆Ψ −
11 + |∇ v | Ψ kj v k v j + W k Ψ k = β −
21 + |∇ v | (cid:26) ( β − (cid:16) µ p |∇ v | (cid:17) + β |∇ v | (cid:27) . (26)If β ∈ [1 , β − ≤ µ > Lemma 3.2. If β ∈ [1 , , then the auxiliary function Ψ ( x ; β ) cannot be identicallyconstant on Ω . roof. If β ∈ [1 , x ; β ) can satisfy (26). Therefore, itremains to investigate the case β = 2. In such a case, if we assume that Ψ ( x ; 2) isconstant on Ω, we may obtain again that v is a radial function, that is v = v ( r ),and Ω is a disk. In radial coordinates, equation (3) can be rewritten as v rr + 1 r v r (1 + v r ) = 1 + v r + µ (1 + v r ) / . (27)On the other hand, since Ψ( x ; 2) is constant, we have that ∂ Ψ /∂r = Ψ ,k u ,k = 0, sothat u rr = (1 + u r ) + µ (1 + u r ) / and (27) thus becomes1 r u r (1 + u r ) = 0 , which is impossible, since u r = 0 for r = 0.We are now ready to prove Theorem 1.1. The proof is obtained again by contradic-tion. Suppose that the minimum of Ψ( x ; β ) is attained at the critical point O of v .We distinguish two cases.1. Case β ∈ (1 , v ( O ) = v ( O ) = v ( O ) = 0, we evaluate(24) and (25) at the origin to findΨ ( O ; β ) = Ψ ( O ; β ) = 0 , respectively Ψ ( O ; β ) = 2 v ( O )1 + µ − βv ( O ) , Ψ ( O ; β ) = 0 , Ψ ( O ; β ) = 2 v µ ( O ) − βv ( O ) . Since Ψ( x ; β ) attains its minimum in O , we have0 ≤ Ψ ( O ; β ) = v ( O ) (cid:18) v ( O )1 + µ − β (cid:19) , ≤ Ψ ( O ; β ) = v ( O ) (cid:18) v ( O )1 + µ − β (cid:19) ≥ . (28)Using now that v ii ( O ) >
0, for i = 1 ,
2, inequalities (28) imply2 v ( O )1 + µ − β ≥ , v ( O )1 + µ − β ≥ . v ( O ) ≥ (1 + µ ) β. (29)On the other hand, evaluating equation (3) at O , we find∆ v ( O ) = 1 + µ. (30)Inserting now (30) into (29) and using the fact that 1 + µ >
0, we concludethat β ≤
1, which contradicts the assumption that β >
1, so that the proof ofTheorem 1.1 is achieved in this case.2. Case β = 1. The same continuity argument employed in the case of problem(1)-(2) in the previous section can be repeated here to show that Theorem 1.1also holds in this case. From Theorem 1.1 we know that Φ( x ; β ) takes its minimum value at some point Q β ∈ ∂ Ω. Here we emphasize the dependence of the points Q β on the parameter β .This implies that 2 α − (cid:0) |∇ u | (cid:1) α − − βu ≥ α − (cid:0) q m (cid:1) α − , (31)where we remind that q m is the minimum value of |∇ u | on ∂ Ω. Evaluating (31) atthe unique minimal point of u , we find − βu min ≥ α − (cid:16)(cid:0) q m (cid:1) α − − (cid:17) . The left hand-side of this inequality is positive and attains its minimum when β = 1,hence − u min ≥ α − (cid:16)(cid:0) q m (cid:1) α − − (cid:17) . (32)Next, we construct a lower bound for q m in terms of the curvature κ ( s ) of ∂ Ω. Let Q = Q . Since Φ( x ; 1) takes its minimum value at Q , we have ∂ Φ( x ; 1) /∂ n ≤ Q , or equivalently, 2 (cid:0) u n (cid:1) α − u n u nn − u n ≤ Q , (33)where u n and u nn are the first and second outward normal derivatives of u on ∂ Ω.As u < u = 0 along ∂ Ω, then u n > u n = |∇ u | on ∂ Ω. Thus theabove inequality (33) becomes2 (cid:0) u n (cid:1) α − u nn ≤ Q . (34)12ow, since the boundary ∂ Ω is smooth, equation (1) can be rewritten in normalcoordinates along ∂ Ω as u nn (1 + u n ) / + κu n (1 + u n ) / = (cid:0) u n (cid:1) − α on ∂ Ω , or equivalently, u nn + κu n (1 + u n ) = (1 + u n ) − α on ∂ Ω . (35)Inserting (35) into (34), we obtain1 ≤ κ ( Q ) q m (1 + q m ) α − ≤ κ ( Q ) (cid:0) q m (cid:1) α +12 where for the last inequality we have used that 2 q m ≤ q m . It then follows that1 κ ( Q ) ≤ (cid:0) q m (cid:1) α +12 . Hence (cid:0) q m (cid:1) α − ≥ (cid:18) κ ( Q ) (cid:19) α − α +1 ≥ (cid:18) κ max (cid:19) α − α +1 , (36)from which inequality (9) follows. Moreover, inserting (36) into (32), we also obtainthe inequality (10) and the proof of Theorem 1.2 is thus achieved. The idea of proof is similar to the one already employed in the previous section toprove Theorem 1.2. From Theorem 1.1 we know that Ψ( x ; 1) takes its minimumvalue at some point Q ∈ ∂ Ω. This implies thatln |∇ v | (cid:16) µ p |∇ v | (cid:17) − v ≥ ln q m (cid:16) µ p q m (cid:17) . (37)Evaluating now (37) at the unique minimal point of v , we obtain − v min ≥ ln (1 + q m ) (1 + µ ) (cid:16) µ p q m (cid:17) . (38)From the facts that ∂ Ψ( x ; 1) /∂ n ≤ v n > ∂ Ω, it follows that2 v nn (1 + v n ) (1 + µ p v n ) ≤ Q . (39)13n the other hand, equation (3) can be rewritten in normal coordinates along ∂ Ω,as v nn (1 + v n ) / + κv n (1 + v n ) / = 1 p v n + µ, or, equivalently, v nn = (1 + v n )(1 − κv n ) + µ (1 + v n ) / , (40)where κ ( s ) is the curvature of ∂ Ω. Inserting now the value of v nn from (40) into(39), we obtain after some simplifications1 κ ( Q ) ≤ q m µ p q m ≤ q m µ , from which inequality (11) follows. Finally, using (11) and the fact that the function f ( x ) = x/ (1 + µx ) is increasing, we easily deduce from (38) the desired inequality(12). For both problems (1)-(2) and (3)-(4), some maximum principles for P -functionshave been already obtained by Payne and Philippin in [19]. We can use them inwhat follows, to derive some upper bound estimates which complement the boundsgiven in Theorem 1.2 and 1.3. − u min From [19, Cor. 1], we know that the function Φ( x ; 2) takes its maximum value atthe (only) critical point of u . This implies1 α − (cid:0) |∇ u | (cid:1) α − − α − ≤ u − u min . Therefore, if α >
1, this inequality leads to |∇ u | ≤ (( α − u − u min ) + 1) α − − . (41)Next, we use inequality (41) to derive an upper bound for − u min . Let P be a pointwhere u = u min and Q be a point on ∂ Ω nearest to P . Let r measure the distancefrom P to Q along the ray connecting P and Q . Clearly we have dudr ≤ |∇ u | . (42)14ntegrating now (42) from P to Q along the ray connecting P to Q , and making useof (41), we obtain I := Z u min du q (1 + ( α −
1) ( u − u min )) α − − ≤ Z QP dr = | P Q | ≤ d, (43)where d is the radius of the largest ball inscribed in Ω. Next, using the substitution v = (1 + ( α −
1) ( u − u min )) − α − , we have I = Z (1 − ( α − u min ) − α − v − α √ − v dv ≥ Z (1 − ( α − u min ) − α − dv √ − v (44)= cos − − ( α − u min ) α − ! , (45)where we have used the fact that v ≤ α >
1, to derive the inequality in (44).From (43) and (45) we obtaincos − − ( α − u min ) α − ! ≤ d. In order to solve this inequality for u min , we must require that d < π/
2. Hence weconclude the following estimate.
Theorem 6.1.
Let d be the radius of the largest ball inscribed in Ω . If α > and d < π/ , then the solution u to problem (1)-(2) satisfies the inequality − u min ≤ α − (cid:18) d ) (cid:19) α − − ! . − v min From [19, Cor. 1], the function Ψ( x ; 2) takes its maximum value at the (only) criticalpoint of v . This implies thatln |∇ v | (cid:18) µ q |∇ v | (cid:19) − ln (cid:18) µ ) (cid:19) ≤ v − v min ,
15o that, after some manipulations, we find(1 + µ ) (cid:0) |∇ v | (cid:1)(cid:18) µ q |∇ v | (cid:19) ≤ e v − v min . (46)Since µ >
0, the left hand-side of (46) is obviously larger than 1 + |∇ v | , so from(46) we are lead to the following inequality |∇ v | ≤ e v − v min ) . (47)Next, following the steps of the previous subsection, from (47), which representsexactly inequality (46) for µ = 0, one may obtain the following result (see also [19],where the case µ = 0 was already investigated): Theorem 6.2.
Let d be the radius of the largest ball inscribed in Ω . If d < π/ ,then the solution v to problem (3)-(4) satisfies the following inequality − v min ≤ ln (cid:18) d ) (cid:19) . (48) We conclude this paper with the following two remarks about the extensions tohigher dimensions and the optimality of the bounds found in this paper.1. The most important ingredient in the proof of Theorem 1.1 is the result aboutthe uniqueness of the critical point of solutions. An extension of our idea of proof tohigher dimension doesn’t work, since in a higher dimension we will have to deal withnodal hypersurfaces instead of nodal lines. One may eventually think at using analternative proof, based on a stronger result, if true, which says that the solutionsto our problems might have convex level sets. In such a case, G.A. Philippin andA. Safoui have proved in [22] that equality sign in (17) can be replaced with theappropriate inequality sign. Unfortunately, as proved by X.-J. Wang in [31], thisconvexity result fails to be true in some particular cases, such as α = 0 in (1) or µ = 0 in (3).2. As for the optimality of our bounds, the equality sign is obtained in our boundestimates from Theorem 1.2 and Theorem 1.3 when the corresponding P-functionsare identically constant. However, in Lemmas 2.4 and 3.2 it was already shown thatthis thing is impossible. Therefore, the bound estimates (9)-(12) are not optimal.As for the bound estimates from Theorems 6.1 and 6.2, the dimension of the space16oesn’t play a role in our computations, so these results still remain true in higherdimension. Moreover, the equality sign in these estimates holds in the limit as Ωdegenerates into a strip region of width 2 d , while µ should also be equal to zero inthe case of the estimate found in Theorem 6.2. References [1] L. Barbu, C. Enache, A minimum principle for a soap film problem in R , Z.Angew. Mth. Phys. 64 (2013), 321–328.[2] M. Bergner, The Dirichlet problem for graphs of prescribed anisotropic meancurvature in R n +1 , Analysis (Munich) 28 (2008), 149–166.[3] S. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976),43–55.[4] C. Enache, Necessary conditions of solvability and isoperimetric estimates forsome Monge-Amp`ere problems in the plane, Proc. Amer. Math. Soc. 143 (2015),309–315.[5] C. Enache, Maximum and minimum principles for a class of Monge-Amp`ereequations in the plane, with applications to surfaces of constant Gauss curva-ture, Comm. Pure Appl. Anal. 13 (2014), 1447–1359.[6] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of SecondOrder. Second edition. Springer-Verlag, Berlin, 1983.[7] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Func.Anal. 13 (2003), 178–215.[8] G. Huisken, C. Sinestrari, Mean curvature flow singularities for mean convexsurfaces, Calc. Var. Partial Differential Equations 8 (1999), 1–14.[9] T. Ilmanen, Elliptic regularization and partial regularity for motion by meancurvature, Mem. Amer. Math. Soc. 108, x+90, 1994.[10] H. Ju, Y. Liu, Dirichlet problem for anisotropic prescribed mean curvatureequation on unbounded domains, J. Math. Anal. Appl. 439 (2016), 709–724.[11] R. L´opez, Invariant surfaces in Euclidean space with a log-linear density, Adv.Math. to appear.[12] R. L´opez, The translating soliton equation, arXiv:1812.00592 [math.DG].1713] X.-N. Ma, A sharp minimum principle for the problem of torsional rigidity, J.Math. Anal. Appl. 233 (1978), 257–265.[14] X.-N. Ma, Sharp size estimates for capillary free surfaces without gravity, PacificJ. Math. 192 (2000), 121–134.[15] T. Marquardt, Remark on the anisotropic prescribed mean curvature equationon arbitrary domain, Math. Z. 264 (2010), 507–511.[16] F. Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853–858.[17] L. Nirenberg, On nonlinear partial differential equations ond H¨older continuity,Comm. Pure Appl. Math. 6 (1953), 103–156.[18] L. E. Payne, G. A. Philippin, Some remarks on the problems of elastic tor-sion and of torsional creep, in Some Aspects of Mechanics of continua, Part I,Jadavpur Univ. Calcutta, India, 1977, 32–40.[19] L. E. Payne, G. A. Philippin, Some maximum principles for nonlinear ellipticequations in divergence form with applications to capillary surfaces and tosurfaces of constant mean curvature, Nonlinear Anal. 3 (1979), 193–211.[20] G. A. Philippin, A minimum principle for the problem of torsional creep, J.Math. Anal. Appl. 68 (1979), 526–535.[21] G. A. Philippin, V. Proytcheva, A minimum principle for the problem of St-Venant in R N , N ≥
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