Local Bounds, Harnack's Inequality and Hölder Continuity for for divergence type elliptic equations with non-standard growth

• Autores: Wolanski, Noemi Irene
• Año de publicación: 2015
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Harnack'S Inequality; Variable Exponent Spaces; Local Bounds.
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard p(x)-type growth. A model equation is the inhomogeneous p(x)-Laplacian. Namely, ∆p(x)u := div |∇u| p(x)−2∇u = f(x) in Ω, for which we prove Harnack’s inequality when f ∈ Lq0 (Ω) if max{1, N p1 } < q0 ≤ ∞. The constant in Harnack’s inequality depends on u only through k|u| p(x)k p2−p1 L1(Ω) . Dependence of the constant on u is known to be necessary in the case of variable p(x). As in previous papers, log-H¨older continuity on the exponent p(x) is assumed. We also prove that weak solutions are locally bounded and H¨older continuous when f ∈ Lq0(x) (Ω) with q0 ∈ C(Ω) and max{1, N p(x) } < q0(x) in Ω. These results are then generalized to elliptic equations div A(x, u, ∇u) = B(x, u, ∇u) with p(x)-type growth.
Fil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina