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Applications of Interpolation theory to the regularity of some equasilinear PDEs

Irshaad Ahmed, Alberto Fiorenza, Maria Rosaria Formica, Amiran Gogatishvili, Abdallah El Hamidi

Abstract

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*} where $Ω$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. Moreover, we collect some well-known and new results for identifying some interpolation spaces and enrich some contents with details.

Applications of Interpolation theory to the regularity of some equasilinear PDEs

Abstract

We present some regularity results on the gradient of the weak or entropic-renormalized solution to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*} where is a bounded smooth domain of , is a nonlinear potential and belongs to non-standard spaces like Lorentz-Zygmund spaces. Moreover, we collect some well-known and new results for identifying some interpolation spaces and enrich some contents with details.

Paper Structure

This paper contains 12 sections, 21 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Definitions of some functional spaces
  3. Interpolation spaces with logarithm function
  4. Preliminaries
  5. Identifications of some interpolation spaces
  6. Interpolation of Hölderian mappings
  7. Estimates of the R² -functional related to the mapping R²
  8. Applications to the regularity of the solution of a R²-Laplacian equation
  9. The quasilinear R²-Laplacian equation
  10. The Hölderian mappings for the case R²
  11. The Hölderian mappings for the case R²
  12. Conclusion

Key Result

theorem 1

(Associate space of an interpolation space). Let $X_0, X_1$ be two Banach function spaces, such that $X_1\subset X_0$. Let $1\leq q<+\infty, \ \alpha\in \mathbb R, \ 0<\theta<1$. Then where $X^{'}_i$ is the associate space of $X_i, \ i=0,1$.

Theorems & Definitions (43)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • theorem 1
  • theorem 2
  • theorem 3
  • proof
  • theorem 4
  • ...and 33 more