Regularity for quasilinear equations on degenerate singular sets

By E. Teixeira. 2012

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We prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy $$\fancyscript{S}(u) := \{X : D u(X) = 0 \}$$S(u):={X:Du(X)=0}. Our main Theorem reveals that along $$\fancyscript{S}(u)$$S(u), $$u$$u is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $$\fancyscript{S}(u)$$S(u), $$Du$$Du enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that $$H^1$$H1-weak solutions to $$\text {div}\left( a_{ij}(X)Du \right) = 0$$divaij(X)Du=0, with $$a_{ij}$$aij elliptic and Dini-continuous are actually $$C^{1,1^{-}}$$C1,1- along $$\fancyscript{S}(u)$$S(u). The results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/133bdcf74b6466225ed1291923a1c7e3a14ece1e

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Citations in the corpus

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Classification of positive D1,p(RN)-solutions to the critical p-Laplace equation in RN☆ (2015, background): “Therefore, for a, A fixed, it follows that wRn is uniformly bounded in L (BA \ Ba) and hence, by [9, 13, 14, 21, 23], it is also uniformly bounded in C(K), 0 < α < 1, for any compact set K ⊂ BA \ Ba.”