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Some minimum principles for a class of nonlinear elliptic problems in divergence form

Cristian Enache, Rafael Lopez

Abstract

In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this convexity property, we develop some minimum principles for an appropriate $P$-function, in the sense of L.~E.~Payne. Finally, this new minimum principle is applied to find a priori estimates for the solutions, in terms of the mean curvature of the boundary of the underlying domain.

Some minimum principles for a class of nonlinear elliptic problems in divergence form

Abstract

In this paper we study a general class of nonlinear elliptic problems in divergence form. First, we prove that the solutions to these problems satisfy a convexity property when the given domain is strictly convex. Then, making use of this convexity property, we develop some minimum principles for an appropriate -function, in the sense of L.~E.~Payne. Finally, this new minimum principle is applied to find a priori estimates for the solutions, in terms of the mean curvature of the boundary of the underlying domain.
Paper Structure (19 sections, 9 theorems, 83 equations)

This paper contains 19 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction and motivation
  2. Geometric background
  3. The equations under study
  4. The general problem
  5. Main results
  6. Organization of the paper
  7. Notation
  8. Proof of Theorem \ref{['t1']}
  9. A constant rank theorem
  10. Concavity in the ball
  11. Proof of Theorem \ref{['t1']}
  12. Proof of Theorem \ref{['t2']}
  13. Proof of Theorem \ref{['t3']}
  14. Proof of Theorem \ref{['t3']}(a)
  15. Proof of Theorem \ref{['t3']}(b)
  16. ...and 4 more sections

Key Result

Theorem 1.2

Assume that $u$ is the solution of problem eq:1.4--eq:1.5 and that $f$ satisfies the following properties: Then $v = v(u)$, defined in eq:1.6--eq:1.6b, is strictly concave in $\Omega$.

Theorems & Definitions (15)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof : Proof for dimension $n=2$
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • ...and 5 more