Quasilinear elliptic problem: Ellipticity-type conditions

Question

Consider the following quasilinear elliptic equation $$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$ on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\partial \Omega} = 0.$$

• Where can I find references on this kinds of problems (in particular about a variational approach to solve them)?
• What's the correct formulation of a "uniform ellipticity"-type condition that gives existence and uniqueness for this problem?

Answer 1

Here is a basic result. For a detailed proof, see Theorem 5.4 in [2].

Theorem. If $\Omega\subset\mathbb{R}^n$ is a bounded Lipschitz domain and $$ F=F(x,u,\xi):\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R} $$ satisfies

• $F$ and $\nabla_\xi F$ are continuous,
• $\xi\mapsto F(x,u,\xi)$ is convex,
• $F(x,u,\xi)\geq c|\xi|^p + a(x)$, $1<p<\infty$, $a\in L^1(\Omega)$,

then for any $w\in W^{1,p}(\Omega)$ the functional $$ I(u)=\int_\Omega F(x,u,\nabla u)\,dx $$ attains minimum in the class $$ W^{1,p}_w(\Omega)=\{ u\in W^{1,p}(\Omega):\, u-w\in W^{1,p}_0(\Omega)\}. $$

Under some additional assumptions about $F$ this minimizer solves the elliptic equation $$ (*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ div A(x,u,\nabla u)=B(x,u,\nabla u), \quad u-w\in W^{1,p}_0(\Omega), $$ where $$ A(x,u,\xi)=\nabla_\xi F(x,u,\xi), \quad B=F'_u(x,u,\xi). $$ Thus one of the method of solving (*) is trough the associated variational problem.

Note also that convexity of $\xi\mapsto F$ implies that $$ (**)\ \ \ \ \ \ \ \ \ \ \langle A(x,u,\xi)-A(x,u,\eta),\xi-\eta\rangle\geq 0 $$ which is known as the monotonicity condition so another method of solving equation (*) satisfying (**) is through the Minty-Browder theorem. The monotonicity condition is a sort of ellipticity.

For an elementary, but detailed presentation of basic variational and non-variational techniques I highly recommend:

[1] L. C. Evans, Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010

Anther elementary presentation is given in:

[2] P. Hajlasz, Non-linear elliptic partial differential equations.. Unpublished lecture notes.

Other books that develop material in depth are:

[3] B. Dacorogna, Direct methods in the calculus of variations. Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.

[4] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. (Great book aut a lot of typos.)

[5] M. Giaquinta, L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 2. Edizioni della Normale, Pisa, 2005.

[6] E. Giusti, Direct Methods in the Calculus of Variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.