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Stable solutions to reaction-diffusion elliptic problems

Xavier Cabre

Abstract

We are concerned with stable solutions to reaction-diffusion elliptic PDEs. We begin with regularity questions, first addressing the classical Laplacian. In joint work with Figalli, Ros-Oton, and Serra, we proved that stable solutions are smooth up to the optimal dimension 9, thereby solving an open problem posed by Brezis in the mid-1990s. We describe this result and also discuss related progress and open problems for the fractional Laplacian -- arising naturally in boundary reaction problems -- , the $p$-Laplacian, and minimal surfaces. We then turn to existence questions, starting with the Casten-Holland and Matano theorem for interior reactions, which states that no nonconstant stable solution exists in convex domains under zero Neumann boundary conditions. We present a recent result with Consul and Kurzke (forthcoming) establishing that the analogous statement fails for boundary reactions. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line, for which we present new results and open problems.

Stable solutions to reaction-diffusion elliptic problems

Abstract

We are concerned with stable solutions to reaction-diffusion elliptic PDEs. We begin with regularity questions, first addressing the classical Laplacian. In joint work with Figalli, Ros-Oton, and Serra, we proved that stable solutions are smooth up to the optimal dimension 9, thereby solving an open problem posed by Brezis in the mid-1990s. We describe this result and also discuss related progress and open problems for the fractional Laplacian -- arising naturally in boundary reaction problems -- , the -Laplacian, and minimal surfaces. We then turn to existence questions, starting with the Casten-Holland and Matano theorem for interior reactions, which states that no nonconstant stable solution exists in convex domains under zero Neumann boundary conditions. We present a recent result with Consul and Kurzke (forthcoming) establishing that the analogous statement fails for boundary reactions. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line, for which we present new results and open problems.
Paper Structure (12 sections, 9 theorems, 51 equations)

This paper contains 12 sections, 9 theorems, 51 equations.

Table of Contents

  1. Hilbert's 19th problem on regularity
  2. Stable minimal surfaces
  3. Regularity of stable solutions to interior reaction problems
  4. The interior regularity result
  5. The boundary result
  6. The three ingredients for the interior estimate
  7. On the radial Poincaré inequality
  8. Related equations. Open problems
  9. Stable solutions to boundary reaction problems
  10. Interior reactions under zero Neumann boundary conditions
  11. Boundary reaction problems
  12. Reaction equations for the fractional Laplacian on the whole real line. Periodic solutions

Key Result

Theorem 3.1

Let $u\in C^\infty(\overline B_1)$ be a stable solution of $-\Delta u=f(u)$ in $B_1\subset \mathbb{R}^n$, for some nonnegative function $f\in C^{1}(\mathbb{R})$. Then, for some dimensional constants $\gamma>0$ and $C$. In addition, where $\alpha>0$ and $C$ are dimensional constants.

Theorems & Definitions (9)

  • Theorem 3.1: CFRS, Theorem 1.2
  • Theorem 3.2: CFRS, Theorem 1.5
  • Proposition 3.3: CFRS, Lemma 2.1
  • Proposition 3.4: CFRS, Proposition 2.5
  • Proposition 3.5: C22quant, Theorem 1.4
  • Theorem 3.6: CSa
  • Theorem 4.1: CastenHolland1978Matano1979
  • Theorem 4.2: CCK
  • Theorem 5.1: CabreCsatoMas2024, Theorem 1.1