THE HYPOELLIPTIC ROBIN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS

. This paper is devoted to the study of a hypoelliptic Robin boundary value problem for quasilinear, second-order elliptic diﬀerential equations depending nonlinearly on the gradient. More precisely, we prove an existence and uniqueness theorem for the quasilinear hypoelliptic Robin problem in the framework of H¨older spaces under the quadratic gradient growth condition on the nonlinear term. The proof is based on the comparison principle for quasilin- ear problems and the Leray–Schauder ﬁxed point theorem. Our result extends earlier theorems due to Nagumo, Akˆo and Schmitt to the hypoelliptic Robin case which includes as particular cases the Dirichlet, Neumann and regular Robin problems.


Introduction and main result.
Let Ω be a bounded domain of the Euclidean space R N , N ≥ 2, with smooth boundary ∂Ω. its closure Ω = Ω ∪ ∂Ω is an Ndimensional, compact smooth manifold with boundary. We consider a second-order, uniformly elliptic differential operator with real smooth coefficients on the closure Ω = Ω ∪ ∂Ω such that (1) a ij (x) = a ji (x) for 1 ≤ i, j ≤ N , and there exists a constant a 0 > 0 such that a ij (x) η i η j ≥ a 0 |η| 2 for all x ∈ Ω and η ∈ R N .
In this paper we study the following quasilinear elliptic boundary value problem with non-homogeneous Robin condition: For a given function ϕ(x ) defined on ∂Ω, find a function u(x) in Ω such that Here: (3) ∇u stands for the gradient of u ∇u = ∂u ∂x 1 , ∂u ∂x 2 , . . . , ∂u ∂x N .
(4) a(x ) and b(x ) are real-valued, smooth functions on the boundary ∂Ω.
(5) n = (n 1 , n 2 , . . . , n N ) is the unit outward normal to ∂Ω. (6) ∂/∂ν is the outward conormal derivative associated with the operator A (see Figure 1)    Moreover, we impose the following two assumptions on the Robin boundary condition B: It should be emphasized that the conditions (H.1) and (H.2) allow the problem (1) to include as particular cases the Dirichlet (a(x ) ≡ 0), Neumann (b(x ) ≡ 0) and regular Robin (a(x ) ≡ 1) boundary conditions. We give a simple but typical example of such functions a(x ) and b(x ) in the case where N = 2 ([22, Example 1.1]): For a local coordinate system x 1 = cos θ, x 2 = sin θ with θ ∈ [0, 2π] on the unit circle ∂Ω, we define a function a(x 1 , x 2 ) by the formula What is the important feature of the conditions (H.1) and (H.2) is that the so-called Lopatinski-Shapiro ellipticity condition is violated at the points where a(x ) = 0 (see [22,Example 6.1]). More precisely, if we reduce the study of the problem (1) to that of a first order, pseudo-differential operator T on the boundary ∂Ω, then the operator T is of the form where ∆ is the Laplace-Beltrami operator on ∂Ω (see [22,Chapter 7]). We can prove that if the conditions (H.1) and (H.2) are satisfied, then the operator T has a parametrix S in the Hörmander class L 0 1,1/2 (∂Ω) (see [22,Lemma 7.2]). Hence the operator T is hypoelliptic with loss of one derivative on ∂Ω.
Remark 1. Amann-Crandall [4] studied the regular (non-degenerate) Robin case. More precisely, they assume that the boundary ∂Ω is the disjoint union of the two subsets M = {x ∈ ∂Ω : a(x ) = 0} and ∂Ω \ M = {x ∈ ∂Ω : a(x ) > 0}, each of which is an (N − 1)-dimensional, compact smooth manifold. In this case, it is easy to see that the pseudo-differential operator T = a(x ) is elliptic of order one on ∂Ω \ M and of order zero on M , respectively.
The linear problem (1), that is, is studied in great detail by Taira [18] and [22] in the frameworks of Hölder and Sobolev spaces. In the case where the function f is nonlinear in u but independent of ∇u, that is, there is a similar result due to Taira [20] where a global static bifurcation theory is elaborated. We remark that Taira [19] studies the homogeneous problem (1) (ϕ ≡ 0) for linear elliptic operators of divergence form by using the super-subsolution method ([8, Section 6.3]). On the other hand, the problem (1) with is related to the so-called Yamabe problem which is a basic problem in Riemannian geometry (see [12], [14], [7], [17]). In this paper the nonlinear term f (x, z, p) of the problem (1) will be subject to the following three conditions: (i) Regularity conditions: f (x, z, p) is continuously differentiable with respect to z and p.
(ii) Monotonicity condition: There exists a constant f 0 > 0 such that (iii) Quadratic gradient growth condition: There exists a positive and nondecreasing function f 1 (t) such that By the quadratic gradient growth condition (4), we find that the nonlinear term f (x, z, p) satisfies the so-called Nagumo condition (see [16, condition (2.6)]): The main purpose of this paper is to extend Taira [19] to the non-homogeneous problem (1) allowing quadratic nonlinearity in f with respect to the gradient ∇u of the unknown function u. We derive an existence and uniqueness result for the problem (1) in the framework of Hölder spaces.
This paper is an expanded and revised version of the previous work Taira-Palagachev-Popivanov [23].
Following Taira [18], we introduce a variant of Hölder space equipped with the norm Then it is easy to verify (see the proof of [22,Lemma 6.8]) that the function space C 1+α * (∂Ω) is a Banach space with respect to the norm · C 1+α * (∂Ω) . We remark that the space C 1+α * (∂Ω) is an "interpolation space" between the Hölder spaces C 2+α (∂Ω) and C 1+α (∂Ω). More precisely, we have the assertions and, for general a(x ), we have the continuous injections Now we are in a position to state our main result: Theorem 1.1. In addition to the conditions (H.1) and (H.2), we assume that the regularity conditions (2), the monotonicity condition (3) and the quadratic gradient growth condition (4) are satisfied. Then the quasilinear problem (1) admits a unique classical solution u ∈ C 2+α (Ω) for any ϕ ∈ C 1+α * (∂Ω).
It should be emphasized that Theorem 1.1 is a generalization of Nagumo [ In this case we may take f 0 = 1 and f 1 (t) = 1 + t.
A typical example of our quasilinear problem (1) is given by the following: in Ω, where (see Example 1) 0 ≤ a(x ) ≤ 1 on ∂Ω.
In this case we may take The rest of this paper is organized as follows. Section 2 is devoted to the precise definitions of Hölder spaces C k+α (Ω), C k+α (Ω) and L p Sobolev spaces W k,p (Ω). In Section 3 we establish a priori estimates of solutions u ∈ C 2+α (Ω) of the nonhomogeneous quasilinear problem (1) (Theorem 3.4). The deriving of the desired a priori estimate (14) is a two-step process consisting of successive bounds on the Hölder norms u C(Ω) and ∇u C α (Ω) in the following way: Section 4 is devoted to the proof of Theorem 1.1. This is carried out by making use of a version of the Leray-Schauder fixed point theorem due to Schaefer (Theorem 4.1) which reduces the solvability of the problem (1) to the establishment of a uniform a priori estimate in the Hölder space C 1+α (Ω) for all solutions of a family of nonlinear problems related to the problem (1) (see the estimate (20)).
In Appendix we formulate various maximum principles due to Bony [5] for secondorder, elliptic differential operators with discontinuous coefficients such as the weak and strong maximum principles (Theorems A.2 and A.4) and the Hopf boundary point lemma (Lemma A.3) in the framework of L p Sobolev spaces.
2. Function spaces. This preparatory section is devoted to the precise definitions of Hölder and Sobolev spaces of L p type (see ).
Let 0 < α < 1. A function u defined on Ω is said to be uniformly Hölder continuous with exponent α in Ω if the quantity is finite. We say that u is locally Hölder continuous with exponent α in Ω if it is uniformly Hölder continuous with exponent α on compact subsets of Ω. If 0 < α < 1, we define the Hölder space C α (Ω) as follows: C α (Ω) = the space of functions in C(Ω) which are locally Hölder continuous with exponent α on Ω.
If k is a positive integer and 0 < α < 1, we define the Hölder space C k+α (Ω) as follows: 1606 KAZUAKI TAIRA C k+α (Ω) = the space of functions in C k (Ω) all of whose k-th order derivatives are locally Hölder continuous with exponent α on Ω.
We introduce various seminorms and norms on the spaces C k (Ω) and C k+α (Ω) as follows: Furthermore, we let Similarly, we define the associated norms on the spaces C k (Ω) and C k+α (Ω) as follows: The usual Sobolev space W k,p (Ω) for k ∈ N and 1 < p < ∞ is defined as follows: W k,p (Ω) = the space of functions u ∈ L p (Ω) whose derivatives D α u, |α| ≤ k, in the sense of distributions are in L p (Ω), and its norm · W k,p (Ω) is given by the formula

3.
A priori estimates for the quasilinear problem (1). In the proof of Theorem 1.1 we make use of a version of the Leray-Schauder fixed point theorem due to Schaefer (Theorem 4.1). For this purpose, we need to establish an a priori estimate for the C 1+α (Ω)-norm of each solution u ∈ C 2+α (Ω) of the non-homogeneous quasilinear problem (1). We start with the following comparison principle for quasilinear problems ([4, Lemma 2]): Lemma 3.1. Assume that the condition (H.1) and (H.2) are satisfied and further that f (x, z, p) is strictly increasing in z for each (x, p) ∈ Ω×R N and is differentiable with respect to p for each ( Then it follows that u(x) ≤ v(x) on Ω.
Proof. Our proof is based on a reduction to absurdity. We let and assume, to the contrary, that the set is non-empty (see Figure 2).   Since f (x, z, p) is strictly increasing with respect to z, it follows from the inequality (5) that However, we can rewrite the term in the form Hence we have, by the inequality (7), Now we take a point x 0 of the closure Ω such that (1) First, we consider the case where x 0 ∈ Ω: We remark that Then it follows from an application of the strong maximum principle (see Theorem Hence we have the inequality This contradicts the inequality (8).

Then it follows from an application of the Hopf boundary point lemma (see Lemma
Hence we have, by conditions (H.1) and (H.2), However, it follows from the inequality (6) that This is a contradiction. Summing up, we have proved that the set Ω + is empty. The proof of Lemma 3.1 is complete.

3.1.
A priori estimate for the uniform norm u C(Ω) . As a first step in obtaining an a priori estimate for the non-homogeneous problem (1), we consider the homogeneous case. Namely, let u ∈ C 2+α (Ω) be a solution of the problem Then we have the following a priori bound on the uniform norm u C(Ω) : Proof. The proof of Lemma 3.2 is divided into two steps.
Step (1). First, by letting we obtain from the monotonicity condition (3) that

Hence it follows that
in Ω.
On the other hand, we have the inequality Therefore, it follows from an application of Lemma 3.1 that Step (2). Secondly, if we let is a solution of the nonlinear problem However, the nonlinear term f (x, z, p) satisfies the monotonicity condition (3): Hence, by arguing just as in Step (1) with u(x) and f (x, z, p) replaced by v(x) and f (x, z, p), respectively, we obtain that since we have the formula The proof of Lemma 3.2 is complete.

3.2.
A priori estimate for the Hölder norm u C 1+α (Ω) . In the following the letter C stands for a generic positive constant depending only on known quantities but not on u, which may vary from a line into another.
We start with an a priori bound on the Hölder norm u C 1+α (Ω) for the homogeneous problem (9): Theorem 3.3. In addition to the conditions (H.1) and (H.2), we assume that the regularity conditions (2), the monotonicity condition (3) and the quadratic gradient growth condition (4) are satisfied. Then there exists a positive constant C, independent of u, such that for every solution u ∈ C 2+α (Ω) of the homogeneous problem Proof. First, it follows from an application of Morrey's lemma (see [ (Ω), α = 1 − N p holds true for p > N . Hence we have, with some constant C > 0, Namely, the a priori bound (11) on the Hölder norm u C 1+α (Ω) can be reduced to a uniform estimate (with respect to u) of the Sobolev norm u W 2,p (Ω) for every solution u of the homogeneous problem (9). However, since the quadratic gradient growth condition (4) is satisfied, we can apply [19, Proposition 2.3] to find a non-negative and increasing function γ(t), depending only on known quantities, such that for every solution u ∈ W 2,p (Ω) of the homogeneous problem (9). Indeed, the proof of [19, Proposition 2.3] remains valid for our operator We remark that the proof of [19, Proposition 2.3] is based on methods developed by Tomi [24] and Amann [3]. Therefore, the desired a priori bound (11) follows by combining the estimates (12), (13) and the a priori bound (10) on the uniform norm u C(Ω) .
The proof of Theorem 3.3 is complete.
The purpose of this subsection is to generalize Theorem 3.3 to the non-homogeneous boundary condition case: Theorem 3.4. In addition to the conditions (H.1) and (H.2), we assume that the regularity conditions (2), the monotonicity condition (3) and the quadratic gradient growth condition (4) are satisfied. Then there exists a positive constant C, independent of u, such that for every solution u ∈ C 2+α (Ω) of the non-homogeneous problem (1) with ϕ ∈ C 1+α * (∂Ω): Here the constant C depends on the norm ϕ C 1+α * (∂Ω) .
Proof. To deal with the non-homogeneous problem (1), we remark that [18, Theorem 1.1] implies the existence of a unique solution v ∈ C 2+α (Ω) of the linear problem Now we introduce a new nonlinear term If u is a solution of the problem (1), then the function solves the homogeneous nonlinear problem It is easy to see that the nonlinear term f (x, z, p) satisfies the monotonicity condition Moreover, we have the inequality However, by the estimate (15) it follows that . Hence we find that the nonlinear term f (x, z, p) satisfies the quadratic gradient growth condition (4) with a new function f 1 (t): |f (x, z, p)| ≤ f 1 (|z|) 1 + |p| 2 for all (x, z, p) ∈ Ω × R × R N .
By applying Theorem 3.3 to the homogeneous problem (16), we obtain from the estimate (11) that with some constant C 2 > 0. Therefore, the desired estimate (14) follows by combining the estimates (15) and (17) with . Indeed, it suffices to note that  The proof of the existence part is divided into four steps.
Step 1. Let ϕ ∈ C 1+α * (∂Ω). For any given function v ∈ C 1+α (Ω), we consider the linear problem In view of the regularity conditions (2), it follows that Hence [18,Theorem 1.1] asserts that there exists a unique solution u ∈ C 2+α (Ω) of the linear problem (18). In this way, we can define a nonlinear operator H by the formula is an algebraic and topological isomorphism for α ∈ (0, 1). This implies the continuity of H considered as an operator from C 1+α (Ω) into C 2+α (Ω). Furthermore, since the space C 2+α (Ω) is compactly imbedded into the space C 1+α (Ω) (see [10,Lemma 6.36]), we derive immediately the compactness of the mapping The situation can be visualized as follows: Step 2. Now, for each ρ ∈ [0, 1] we consider the equation that is, the non-homogeneous problem We shall prove the following uniform a priori estimate for every solution u = u ρ of the non-homogeneous problem (19) with a constant C > 0 independent of ρ and u.
If v ∈ C 2+α (Ω) is a unique solution of the linear problem Then it follows that v ρ is the unique solution of the linear problem with the estimate for all 0 ≤ ρ ≤ 1.
Here it should be noticed (see the estimate (14)) that the constant C 1 depends on the norm ϕ C 1+α * (∂Ω) . Substep 2.2. For every solution u = u ρ of the non-homogeneous problem (19), we let Then it follows from the problems (19) and (21) that the function w ρ is a unique solution of the homogeneous nonlinear problem We remark that w 0 = 0 for ρ = 0 as it follows from the uniqueness result in [18,Theorem 1.1]. Therefore, if we introduce a new nonlinear term then the non-homogeneous nonlinear problem (23) can be expressed as follows: We verify that the nonlinear term f ρ (x, z, p) satisfies the monotonicity condition (3). Indeed, it follows that On the other hand, it follows from the estimate (22) that By applying Lemma 3.2 and Theorem 3.3 to the homogeneous problem (24), we obtain the uniform estimate with some constant C 3 > 0 independent of ρ. We recall that w 0 = 0 for ρ = 0.
Substep 2.3. Moreover, we have the inequality Hence we find that the nonlinear term f ρ (x, z, p) satisfies the quadratic gradient growth condition (4) with a new function f 1 (t): Therefore, the desired estimate (20) follows by combining the estimates (22) and (25) with . Indeed, it suffices to note that We remark that u 0 = v 0 + w 0 = 0 for ρ = 0.
Step 3. By using Schaefer's theorem (Theorem 4.1), we find that the properties of the operator H and the estimate (20) imply the existence of a fixed point u ∈ C 1+α (Ω) of the operator H. Namely, the function u satisfies the non-homogeneous problem (19) for ρ = 1: In this way, the fixed point u becomes a solution of the original nonlinear problem (1).
Step 4. Finally, the smoothing properties of H yield that u = Hu ∈ C 2+α (Ω). Now the proof of Theorem 1.1 is complete.
Appendix: The maximum principle in Sobolev spaces. In this appendix we formulate various maximum principles for second-order, elliptic differential operators with discontinuous coefficients such as the weak and strong maximum principles (Theorems A.2 and A.4) and the boundary point lemma (Lemma A.3) in the framework of L p Sobolev spaces. The results here are adapted from Bony [5], Troianiello [25,Chapter 3] and also Taira [21,Chapter 8].
Let Ω be a bounded domain in Euclidean space R N , N ≥ 3, with boundary ∂Ω of class C 1,1 . We consider a second-order, uniformly elliptic differential operator A with real discontinuous coefficients of the form More precisely, we assume that the coefficients a ij (x), b i (x) and c(x) of the differential operator A satisfy the following three conditions: (1) a ij (x) ∈ L ∞ (Ω), a ij (x) = a ji (x) for almost all x ∈ Ω and there exist a constant λ > 0 such that a ij (x)ξ i ξ j ≤ λ|ξ| 2 for almost all x ∈ Ω and all ξ ∈ R N .
First, we state a variant of the weak maximum principle in the framework of L p Sobolev spaces, due to Bony [5]  If u(x) attains a non-negative, strict local maximum at a point x 0 of ∂Ω, then we have the inequality ∂u ∂ν (x 0 ) > 0 (see Figure 1).
If u(x) attains a non-negative maximum at an interior point x 0 of Ω, then it is a (non-negative) constant function.