SOLUTIONS TO GENERAL QUASILINEAR ELLIPTIC SECOND ORDER PROBLEMS

P. AMSTER, M.M. CASSINELLI AND M. C. MARIANI

Abstract. We study the second order quasilinear equation (1) Qu = f(x, u, D_{1}u, ..., D_{n}u) with Dirichlet conditions. We obtain existence and uniqueness results in the Sobolev Space W^{2,p}. We also prove that, in certain cases, the set of solutions of (1) is a connected subset of W^{2,p}.

1. Statement of the problem and known results

We consider the Dirichlet problem in a bounded C^{1,1} domain Ω ⊂ R^{n} for u : \overline{Ω} ⟶ R satisfying

(2) \begin{cases}Qu = f(x, u, D_{1}u, ..., D_{n}u) in Ω \\ u = g in ∂Ω \end{cases}

where Q is the quasilinear elliptic operator given by Qu = ∑ a_{ij}D_{ij}u + ∑ b_{j}D_{j}u + cu, with a_{ij}(x, u, D_{1}u, ..., D_{n}u) continuous and C^{1} in u, D_{1}u, ..., D_{n}u, b_{j}(x, u) continuous and C^{1} in u, and c ∈ L^{∞}(Ω). We’ll assume that Q is strictly elliptic, i.e. if λ_{i} and λ_{s} are respectively the minimum and the maximum eigenvalues of the matrix a_{ij}(x, u, D_{1}u, ..., D_{n}u), then λ_{i} ⩾ λ > 0. We’ll assume also that f : \overline{Ω} × R × R^{n} ⟶ R is continuous and g ∈ W^{2,p}(Ω) for some n < p < ∞.

Existence results for (1) have been obtained in certain cases. For example, for Q uniformly elliptic, i.e. such that λ_{s}/λ_{i} is bounded, see [?]. A well known example of a non-uniformly elliptic problem is the mean curvature equation for non parametric surfaces (here λ_{i} = 1, λ_{s} = 1+|∇u|^{2}) which has been studied by Gilbarg, Trudinger, Simon, Serrin, among other authors [?], [?], [?], [?]. A more general situation is contemplated in [?],[?],[?].

We linearize the problem in the following way: for fixed \overline{u} ∈ C^{1}(\overline{Ω}), we consider the linear elliptic operator

L\overline{u}v = ∑ a_{ij}(x, \overline{u}, D_{1}\overline{u}, ..., D_{n}\overline{u})D_{ij}v + ∑ b_{j}(x, \overline{u})D_{j}v + cv

For c ⩽ 0 we recall the apriori bound for L\overline{u} [?]: there exists a constant c = c(\overline{u}) such that

∥w∥_{2,p} ⩽ c∥L\overline{u}w∥_{p}

for any w ∈ W^{2,p} ∩ W_{0}^{1,p}. It is immediate that we can choose c(\overline{u}) minimum, then we obtain the following result:

Lemma 1.1. c : C^{1} ⟶ R^{+} is upper semicontinuous.

1991 Mathematics Subject Classification. Primary 35H30 ; Secondary 35J40.

Key words and phrases. nonlinear problems, fixed points methods.

This work was partially supported by UBACYT TX45.

Proof. Let t > c(\overline{u}), w ∈ W^{2,p} ∩ W_{0}^{1,p}. For ∥\overline{v} − \overline{u}∥_{1,∞} ⩽ R, we have:

∥L\overline{u}w∥_{p} ⩾ \frac{1}{c(\overline{u})}∥w∥_{2,p} − ∥(L\overline{u} − L\overline{v})w∥_{p}

As

∥(L\overline{u} − L\overline{v})w∥_{p} ⩽ ∥ ∑(a_{ij}(x, \overline{u}, D_{1}\overline{u}, ..., D_{n}\overline{u}) − a_{ij}(x, \overline{v}, D_{1}\overline{v}, ..., D_{n}\overline{v}))D_{ij}w+

∑(b_{j}(x, \overline{u}) − b_{j}(x, \overline{v}))D_{j}w∥_{p} ⩽ kR∥w∥_{2,p}

for some constant k, we obtain:

∥L\overline{u}w∥_{p} ⩾ (\frac{1}{c(\overline{u})} − kR)∥w∥_{2,p}

and choosing R such that kR < \frac{1}{c(\overline{u})} it follows that \frac{1}{c(\overline{v})} ⩾ \frac{1−kRc(\overline{u})}{c(\overline{u})}. Letting

R ⟶ 0 we conclude that c(\overline{v}) ⩽ \frac{c(\overline{u})}{1−kRc(\overline{u})} < t and the result holds.

Remark 1.1. In the situation of lemma 1.1, if kR < \frac{1}{c(\overline{u})} then it is possible to fix a constant \overline{c} such that \overline{c} ⩾ c(\overline{v}) for any \overline{v} ∈ B_{R}(\overline{u}). We also remark that for a bounded set C ⊂ W^{2,p} ↪ C^{1} the constant c(\overline{u}) may be chosen uniformly in C.

In order to solve (2), we’ll apply a coefficient freezing method that will allow us to define an operator T : C^{1}(\overline{Ω}) ⟶ C^{1}(\overline{Ω}) and find solutions by a fixed point theorem. Indeed, for \overline{u} ∈ C^{1} we call T \overline{u} the unique solution in W^{2,p} ↪ C^{1} of the linear problem

\begin{cases}L\overline{u}u = f(x, \overline{u}, D_{1}\overline{u}, ..., D_{n}\overline{u}) in Ω \\ u = g in ∂Ω \end{cases}

Lemma 1.2. T is continuous and locally compact.

Proof. Let \overline{u} ∈ C^{1} and R > 0 such that kR < \frac{1}{c(\overline{u})} where k is the constant of lemma 1.1. Fixing \overline{c} such that \overline{c} ⩾ c(\overline{v}) for any \overline{v} ∈ B_{R}(\overline{u}), we obtain:

∥T \overline{u} − T \overline{v}∥_{1,∞} ⩽ c_{1}∥T \overline{u} − T \overline{v}∥_{2,p} ⩽ \overline{c}c_{1}∥L\overline{v}(T \overline{u} − T \overline{v})∥_{p} ⩽
\overline{c}c_{1}(∥L\overline{u}T \overline{u} − L\overline{v}T \overline{u}∥_{p} + ∥(L\overline{u} − L\overline{v})T \overline{u}∥_{p}) ⩽

\overline{c}c_{1}(∥f(x, \overline{u}, D_{1}\overline{u}, ..., D_{n}\overline{u}) − f(x, \overline{v}, D_{1}\overline{v}, ..., D_{n}\overline{v})∥_{p} + k∥T \overline{u}∥_{2,p}∥\overline{u} − \overline{v}∥_{1,∞}). Then continuity follows. Furthermore, as T (B_{R}(\overline{u})) ⊂ W^{2,p} is bounded, by

the compactness of the imbedding W^{2,p} ↪ C^{1}, we conclude that the closure of T (B_{R}(\overline{u})) is compact.

The next result establishes the existence of a solution in a neighborhood of any u satisfying Qu = f(x, u, D_{1}u, ..., D_{n}u), if the boundary data g is close enough to u. More generally, we can also allow small perturbations of f:

Theorem 1.3. Let g_{0} ∈ W^{2,p}, u_{0} ∈ W^{2,∞} satisfy Qu_{0} = f_{0}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) in Ω, u_{0} = g_{0} on ∂Ω. We define

ψ(x) := ∑_{i,j} \frac{∂a_{ij}}{∂u} (x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{ij}u_{0} + ∑_{j} \frac{∂b_{j}}{∂u} (x, u_{0})D_{j}u_{0}

and assume that one of the following conditions holds:

i) f_{0} is Lipschitz in u, D_{1}u, ..., D_{n}u with constant small enough and ψ + c ⩽ 0. ii) f_{0} is C^{1} in u, D_{1}u, ..., D_{n}u and ψ + c ⩽ \frac{∂f_{0}}{∂u} (·, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}).

Then problem (2) is solvable in W^{2,p} for any (f, g) close enough to (f_{0}, g_{0}) in C(\overline{Ω} × R × R^{n}) × W^{2,p}(Ω).

Proof. Let (f, g) be in a neighborhood of (f_{0}, g_{0}). We look for a solution of (2) for (f, g), which is equivalent to find u ∈ W^{2,p}(Ω) such that u = g in ∂Ω and

L_{u}u − L_{u_{0}}u_{0} = f(x, u, D_{1}u, ..., D_{n}u) − f(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) in Ω

For z = u − u_{0}, we have that

L_{u}u − L_{u_{0}}u_{0} = ∑ a_{ij}(x, u, D_{1}u, ..., D_{n}u)D_{ij}z + ∑ b_{j}(x, u)D_{j}z + cz+

∑(a_{ij}(x, u, D_{1}u, ..., D_{n}u)−a_{ij}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}))D_{ij}u_{0}+∑(b_{j}(x, u)−b_{j}(x, u_{0}))D_{j}u_{0}

We may write

a_{ij}(x, u, D_{1}u, ..., D_{n}u) − a_{ij}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) = \frac{∂a_{ij}}{∂u} (x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})z

+ ∑_{k} \frac{∂a_{ij}}{∂D_{k}u}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{k}z − ρ_{ij}(x, z, D_{1}z, ..., D_{n}z)

b_{j}(x, u) − b_{j}(x, u_{0}) = \frac{∂b_{j}}{∂u} (x, u_{0}) − ρ_{j}(x, z)

and define the quasilinear operator

Mz = ∑ a_{ij}(x, z + u_{0}, D_{1}(z + u_{0}), ..., D_{n}(z + u_{0}))D_{ij}z+

∑_{k} (b_{k}(x, z + u_{0}) + ∑_{i,j} \frac{∂a_{ij}}{∂D_{k}u}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{ij}u_{0}) D_{k}z+

(∑_{i,j} \frac{∂a_{ij}}{∂u} (x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{ij}u_{0} + ∑_{j} \frac{∂b_{j}}{∂u} (x, u_{0})D_{j}u_{0} + c) z

Thus, the problem becomes

\begin{cases}Mz = ∑ ρ_{ij}D_{ij}u_{0} + ∑ ρ_{j}D_{j}u_{0} + F in Ω \\ z = g − g_{0} in ∂Ω \end{cases}

where F (x, z, D_{1}z, ..., D_{n}z) = f(x, u, D_{1}u, ..., D_{n}u) − f_{0}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}). As the operator M satisfies the required conditions, lemma 1.1 holds, and we may define a continuous and locally compact operator T : \overline{z} ⟶ z as in lemma 1.2. Then, for R small we fix \overline{c} such that \overline{c} ⩾ c(\overline{z}) for any \overline{z} ∈ B_{R}(0) ⊂ C^{1}. If i) holds, we obtain:

∥T (\overline{z})∥_{1,∞} ⩽ ∥g − g_{0}∥_{1,∞} + ∥T (\overline{z}) − (g − g_{0})∥_{1,∞} ⩽
∥g − g_{0}∥_{1,∞} + \overline{c}c_{1}∥L\overline{z}(T (\overline{z}) − (g − g_{0}))∥_{p} ⩽ ∥g − g_{0}∥_{1,∞}+

\overline{c}c_{1}(∥ ∑ ρ_{ij}D_{ij}(u_{0}) + ∑ ρ_{j}D_{j}(u_{0}) + F (x, \overline{z}, D_{1}\overline{z}, ..., D_{n}\overline{z})∥_{p} + ∥L\overline{z}(g − g_{0}))∥_{p}).

Moreover,

∥F (x, \overline{z}, D_{1}\overline{z}, ..., D_{n}\overline{z})∥_{p} ⩽

∥f(x, \overline{z} +u_{0}, D_{1}(\overline{z} +u_{0}), ..., D_{n}(\overline{z} +u_{0}))−f_{0}(x, \overline{z} +u_{0}, D_{1}(\overline{z} +u_{0}), ..., D_{n}(\overline{z} +u_{0}))∥_{p}+

∥f_{0}(x, \overline{z} + u_{0}, D_{1}(\overline{z} + u_{0}), ..., D_{n}(\overline{z} + u_{0})) − f_{0}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})∥_{p} and then taking (f, g) close enough to (f_{0}, g_{0}) it’s easy to conclude that

∥T (\overline{z})∥_{1,∞} ⩽ R

for some R small enough. Hence, T (B_{R}(0)) ⊂ B_{R}(0) and the result follows from Schauder fixed point theorem.

Under assumption ii), the proof is similar, writing

f_{0}(x, u, D_{1}u, ..., D_{n}u) − f_{0}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) = \frac{∂f_{0}}{∂u} (x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})z

+ ∑_{k} \frac{∂f_{0}}{∂D_{k}u}(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{k}z + ρ(x, u, D_{1}u, ..., D_{n}u)

and defining the operator

M′z = Mz − \frac{∂f_{0}}{∂u} (·, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})z − ∑_{k} \frac{∂f_{0}}{∂D_{k}u}(·, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})D_{k}z.

Remark 1.2. a) In i) and ii) the condition on ψ may be replaced by the assumption that ∥ψ + c∥_{p} (respectively ∥ψ + c − \frac{∂f_{0}}{∂u} (·, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0})∥_{p}) is small enough.

b) Theorem 1.3 can be combined with the non-existence conditions in [?] (see e.g. theorem 14.12, and corollary 14.13 for the mean curvature equation). For instance, if H > 0 satisfies the existence conditions of [?], theorem 16.10, with H′(y) = \frac{n}{n−1}H(y) at some point y ∈ ∂Ω where H′ is the mean curvature of ∂Ω, there is a sequence H_{n} ⟶ H satisfying the conditions of [?], corollary 14.13. Thus, there exists a sequence g_{n} such that ∥g_{n}∥_{∞} ⟶ 0 and the Dirichlet problem is not solvable. From theorem 1.3 we conclude that no subsequence of g_{n} converges in W^{2,p}.

In the same way we can prove the local uniqueness of the solutions of (2). More precisely:

Theorem 1.4. Let u_{0} ∈ W^{2,∞} be a solution of (2), ψ as in the previous theorem and assume that one of the following conditions holds:

i) f is Lipschitz in u, D_{1}u, ..., D_{n}u with constant small enough and ψ + c ⩽ 0. ii) f is C^{1} in u, D_{1}u, ..., D_{n}u and ψ + c ⩽ \frac{∂f}{∂u}(·, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}).

Then u_{0} is isolated in C^{1}.

Proof. As in the previous theorem, for f = f_{0} and g = g_{0}, we obtain for small R and \overline{z} ≠ 0 that

∥T (\overline{z})∥_{1,∞} < ∥\overline{z}∥_{1,∞}

and the result holds.

Furthermore, in certain cases we can prove the global uniqueness:

Theorem 1.5. Let u_{0} ∈ W^{2,∞} be a solution of (2) and assume that a_{ij}, b_{j} do not depend of u and f is C^{1} in u, D_{1}u, ..., D_{n}u. Then, if c ⩽ \frac{∂f}{∂u}(y) for any y ∈ R^{n+1}, u_{0} is unique in C^{1}.

Proof. As in theorem 1.3, if u is another solution of (2), we have that L_{u}u−L_{u_{0}}u_{0} = f(x, u, D_{1}u, ..., D_{n}u) − f(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) and u = u_{0} on ∂Ω. We can choose φ_{ij}^{k}, φ, φ^{k} ∈ L^{∞} such that

a_{ij}(x, D_{1}u, ..., D_{n}u) − a_{ij}(x, D_{1}u_{0}, ..., D_{n}u_{0}) = ∑_{k} \frac{∂a_{ij}}{∂D_{k}u}(x, φ_{ij}^{1}, ..., φ_{ij}^{n})D_{k}z

and

f(x, u, D_{1}u, ..., D_{n}u) − f(x, u_{0}, D_{1}u_{0}, ..., D_{n}u_{0}) = \frac{∂f}{∂u} (x, φ, φ^{1}, ..., φ^{n})z

+ ∑_{k} \frac{∂f}{∂D_{k}u}(x, φ^{1}, ..., φ^{n})D_{k}z

where z = u − u_{0}. Thus, we obtain:

Lz = 0 in Ω z = 0 in ∂Ω

for some linear elliptic operator satisfying the conditions of lemma 9.17 in [?] and the result follows.

2. Regularity of the solutions

Theorem 2.1. Let u ∈ W^{2,p}(Ω) be a solution of (2), and assume for some k ⩾ 0, 0 < α < 1 that a_{ij}, b_{j}, f ∈ C^{k,α}, ∂Ω ∈ C^{k+2,β}, c ∈ C^{k,β} and g ∈ C^{k+2,β} where

β = α(1 − \frac{n}{p}). Then u ∈ C^{k+2,β}(\overline{Ω}).

Proof. Case k = 0: by Sobolev immersion u ∈ C^{1,1−n/p}(\overline{Ω}). Then Lu ∈ C^{β}(\overline{Ω}) where the elliptic operator L has C^{β} coefficients. By theorem 6.14 in [?], the equation Lw = f en Ω, w = g en ∂Ω is uniquely solvable in C^{2,β}(\overline{Ω}), and the result follows from the uniqueness in theorem 9.15 in [?].

The general case is now immediate, from theorem 6.19 in [?].

3. Properties of the set {Qu = f}

Let us assume that ∂Ω ∈ C^{2,β}, f ∈ C^{α}, and that f and Q satisfy the hypothesis of theorem 1.5.

We consider the sets:

M_{f} = {u ∈ W^{2,p}(Ω)/Qu = f(x, u, D_{1}u, ..., D_{n}u) in Ω}
S_{f} = {g ∈ W^{2,p}(Ω) harmonic / (2) has a solution in W^{2,p}(Ω)}

M_{f} is closed W^{2,p}(Ω, R), since M_{f} = F^{−1}(0) for some F : W^{2,p}(Ω, R) ⟶ L^{p} continuous. Moreover, by theorem 1.3 and 2.1 S_{f} is open in the closed subspace Δ^{−1}(0) ⊂ W^{2,p}(Ω). By theorems 1.5 and 2.1 we may define a one to one map λ : M_{f} ∩ C^{2,β}(\overline{Ω}) ⟶ S_{f} ∩ C^{2,β}(\overline{Ω}), where λ(u) is the only harmonic g such that u = g in ∂Ω. It’s immediate to see that λ is continuous, since

∥λ(u)−λ(v)∥_{2,p} ⩽ ∥λ(u)−u−(λ(v)−v)∥_{2,p}+∥u−v∥_{2,p} ⩽ c∥Δu−Δv∥_{2,p}+∥u−v∥_{2,p}.

Actually, λ may be thought as the restriction of a linear operator \overline{λ} : W^{2,p}(Ω) ⟶ Δ^{−1}(0) which is onto and continuous (and hence, open).

In order to see that λ is an homeomorfism, it suffices to consider a sequence g_{n} ⟶ g, and note that if λ(u) = g, we may find, following the proof of theorem 1.3, for every n big enough a solution \overline{u}_{n} ∈ B_{R_{n}}(u) of the problem with boundary data g_{n}. By uniqueness, \overline{u}_{n} = u_{n}. Moreover, with the notations of theorem 1.3 ii) for g = g_{n} and f = f_{0} we have that if \overline{z} ∈ B_{R_{n}}(0) then

∥T′(\overline{z})∥_{1,∞} ⩽ k∥g_{n} − g_{0}∥_{2,p} + c_{1}c∥ρ∥_{p}

As c_{1}c∥ρ∥_{p} is independent of g_{n} it’s clear that we may take R_{n} ⟶ 0 when g_{n}−g ⟶ 0.

As a simple consequence we have the following

Theorem 3.1. Let us assume that ∂Ω ∈ C^{2,β}, and that Q, f ∈ C^{α} satisfy the hypothesis of theorem 1.5.

Then, if the problem (1) admits a solution for any g, (M_{f} ∩ C^{2,β}(\overline{Ω}), ∥ · ∥_{2,p}) is homeomorphic to the dense subspace C^{2,β} ∩ Δ^{−1}(0) ⊂ (Δ^{−1}(0), ∥ · ∥_{2,p}). In particular, it is connected by arcs.

Remark 3.1. Theorem 3.1 holds for the solutions of the mean curvature equation for nonparametric surfaces with constant H, if the curvature H′ of ∂Ω verifies the condition H′ ⩾ \frac{n|H|}{n−1} (see [?], corollary 16.11).

References

[1] Amster P. Mariani, M.C, Rial, D.F: Existence and uniqueness of H-System’s solutions with Dirichlet conditions. To appear in Nonlinear Analysis, Theory, Methods, and Applications.

[2] Díaz J., Saa J., Thiel U: Sobre la ecuación de curvatura media prescripta y otras ecuaciones cuasilineales elípticas con soluciones anulándose localmente, Revista de la Unión Matemática Argentina vol.35, 1989, 175-206.

[3] Giaquinta, M.: On the Dirichlet problem for surfaces of prescribed mean curvature. Manuscripta Math. 12, 73/86 (1974).

[4] Gilbarg, D. Trudinger, N. S. : Elliptic partial differential equations of second order, Springer-Verlag (1983).

[5] Serrin, J.: Gradient estimates for solutions of nonlinear elliptic and parabolic equations. In: Contributions to Nonlinear Functional Analysis, pp 565-601. New York, Academic Press (1971).

[6] Simon, L.: Equations of mean curvature type in 2 independent variables. Pacific J.Math (1977).

[7] Wang Guofang: The Dirichlet problem for the equation of prescribed mean curvature, Analyse Nonlinéaire 9 (1992), 643-655.

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, ARGENTINA

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E-mail address: pamster@dm.uba.ar --- mcmarian@dm.uba.ar