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In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders
Ω = {(x0,x,t) ∈ R x Rn-1 x R: x0 > A(x,t)}.
We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp
...
Let be a bounded simply connected domain in the complex plane, . Let be a neighborhood of , let be fixed, and let be a positive weak solution to the Laplace equation in Assume that has zero boundary values on in the Sobolev sense and extend to by putting on Then there exists a positive finite Borel measure on with support contained in and such that
whenever If and if is the Green function for with pole at then the measure coincides...
Let , and let , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic measure...
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