Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topics in elliptic problems: from semilinear equations to shape optimization

Hugo Tavares

Abstract

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.

Topics in elliptic problems: from semilinear equations to shape optimization

Abstract

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.
Paper Structure (24 sections, 28 theorems, 120 equations, 3 figures)

This paper contains 24 sections, 28 theorems, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Semilinear elliptic problems: old and new
  3. Dirichlet boundary conditions: the linear case $p=1$.
  4. Dirichlet boundary conditions: the sublinear case $0<p<1$.
  5. Dirichlet boundary conditions: the superlinear--subcritical case $1<p<2^*-1$.
  6. Dirichlet boundary conditions: the critical case $p=2^*-1=(N+2)/(N-2)$
  7. Lane-Emden equations with Neumann boundary conditions
  8. Elliptic Hamiltonian systems
  9. Dirichlet boundary conditions: least energy nodal solutions
  10. Neumann boundary conditions: least energy solutions in the subcritical and critical cases
  11. The subcritical case.
  12. The critical case.
  13. Existence of fully nontrivial solutions to a class of gradient elliptic systems
  14. Subcritical case
  15. Cooperative case
  16. ...and 9 more sections

Key Result

theorem 1

Let $0<p<1$. There exist $u\in H^1_0(\Omega)\cap L^\infty(\Omega)$ such that $\mathcal{I}(u)=c_{nod}$. Moreover, any function achieving the level $c_{nod}$ is a least energy nodal solution.

Figures (3)

  • Figure 1: The curve on top is the critical hyperbola. The hyperbola below corresponds to $pq=1$. Image taken from BonheureSantosTavares.
  • Figure 2: Examples of the functions $\mathcal{I} h$, $\mathfrak{F} h$ and $h^\divideontimes$ for a particular radial function $h\in C(\overline{B_1(0)})$. Images taken from SaldanaTavares.
  • Figure 3: Example of a regular 5-partition in dimension $N=2$.

Theorems & Definitions (32)

  • theorem 1: BSPTW
  • theorem 2: Mountain Pass Theorem
  • theorem 3: Combination of PariniWeth2015CK91 with my recent papers PistoiaSchieraTavaresSaldanaTavares
  • theorem 4: PistoiaSaldanaTavares
  • theorem 5: BMRT15
  • corollary 1: BMRT15
  • theorem 6: SaldanaTavares
  • definition 1: SaldanaTavares
  • theorem 7
  • theorem 8: PistoiaSchieraTavares
  • ...and 22 more