Regularity of solutions to a class of variable–exponent fully nonlinear elliptic equations

By Anne C. Bronzi, Edgard A. Pimentel, G. Rampasso, E. Teixeira. Journal of Functional Analysis, 2018

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In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the $p(x)$-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class $C^{1,\kappa}$ for a universal constant $0< \kappa < 1$. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.

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

View Paper: https://www.semanticscholar.org/paper/d18877e34759458995b6cf25bddc1299ab8abf05

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Recent developments in problems with nonstandard growth and nonuniform ellipticity (2021, background): “, |Dw|p(x)F (D2u) ∈ L∞, has been instead considered in [36].”