Boundary Weak Harnack Estimates and Regularity for Elliptic Operators in Divergence Form and Applications in PDEs

We obtain a global extension of the classical Weak Harnack Inequality which extends and quanties the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form, under the weakest assumptions on the leading coefficients and on the boundary of the domain. Our main tool is the use of suitable barrier functions, which are solutions of auxiliary problems and the C1-estimates up to the boundary.

Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma. Furthermore, we provide
an application showing how to use this results in order to deduce a priori upper bounds and multiplicity of solutions for a class of quasilinear elliptic problems with quadratic growth on the gradient.

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Meeting ID: 899 4652 5336