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Unified A-priori Estimates for Minimizers under $p$, $q$-Growth and Exponential Growth

Dra. Cintia Pacchiano Camacho

IM-Cuernavaca, UNAM

26 de febrero de 2026. 11:30 hrs. Auditorio José Ádem.

Unirse a Zoom
ID de la reunión: 840 6072 8676
Contraseña: 637374

Resumen: In this talk, we present a unified regularity framework for variational integrals with non-uniformly elliptic integrands, including those exhibiting $p$, $q$-growth or exponential-type growth. We consider general energy functionals of the form

$$\int_{\Omega} f(x,Du) dx,$$

where the integrand $f(x, ξ)$ may satisfy natural growth, ($p, q$)-growth, or exponential growth conditions. We establish that under suitable structural assumptions on the second derivatives of $f$ with respect to the gradient variable, any local minimizer is locally Lipschitz continuous. This key result allows us to reduce complex non-uniformly elliptic problems to a standard growth setting, where classical regularity theory can be applied.

Our analysis includes models beyond the uniformly elliptic case, such as anisotropic energies, the double phase functional, the $p(x)$-Laplacian, and exponential growth integrals. We show that a-priori estimates on the gradient and second derivatives of minimizers can be derived under general conditions involving functions $g_1$, $g_2$, and $g_3$ controlling the ellipticity and the regularity of $f$. These estimates serve as a crucial step in proving higher regularity
results.

The results presented extend and unify various existing regularity theories and provide new insights, particularly for variational problems where the integrand grows faster than any polynomial at infinity. We conclude with examples demonstrating the applicability of our theory in multiple settings, including degenerate, anisotropic, and exponential-type energies.