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Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth

Paolo Marcellini, Antonella Nastasi, Cintia Pacchiano Camacho

Abstract

We propose some general growth conditions on the function $% f=f\left( x,ξ\right) $, including the so-called natural growth, or polynomial, or $p,q-$growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral $\;\int_{Ω}f\left( x,Du\right) dx\,$ is locally Lipschitz continuous in $Ω$. In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand $f\left( x,ξ\right) $ as $\left\vert ξ\right\vert \rightarrow +\infty $; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.

Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth

Abstract

We propose some general growth conditions on the function , including the so-called natural growth, or polynomial, or growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral is locally Lipschitz continuous in . In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand as ; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.

Paper Structure

This paper contains 23 sections, 3 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Consequences and examples
  4. Anisotropic energy integrals
  5. Exponential growth
  6. $p(x)-$Laplacian and logarithm $p(x)-$Laplacian
  7. Double phase case
  8. Proof of Theorem \ref{['Lipschitz continuity result']}
  9. Step 1. Second variation
  10. Terms to estimate from 1 to 6
  11. Step 2. Main estimates
  12. Estimate of the term in $\fbox{1}$
  13. Use of ellipticity in the term in $\fbox{2}$
  14. Use of ellipticity in the term in $\fbox{3}$
  15. Estimate of the term in $\fbox{4}$
  16. ...and 8 more sections

Key Result

Theorem 2.2

Under the ellipticity and growth conditions (ellipticity)-(condition on beta), let $u$ be a local minimizer of the energy integral (energy integral) in the Sobolev class (a priori smooth class for local minimizers). Then some uniform a-priori estimates hold for the $L_{\mathrm{loc}}^{\infty }-$norm for every $\rho$,$R$ with $0<\rho <R\leq R_{0}$and for a positive constant $c$, where $B_{\rho }$,

Theorems & Definitions (10)

  • Remark 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Example 3.3: Lipschitz continuity result for anisotropic energy inte-grals
  • Remark 3.4
  • Example 3.5: Lipschitz continuity result under exponential growth
  • Example 3.6: Lipschitz continuity for $p(x)$ Laplacian integral
  • Example 3.7
  • Remark 4.1