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Register a new userOblique Derivative Problems for Elliptic Equations on Conical Domains
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Matthew R. I. Schrecker
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We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Holder regularity of the gradient to hold up to the vertex of the cone. The proof of regularity is based on the application of carefully constructed barrier methods or via perturbative arguments. In the case that such regularity does not hold, we give explicit counterexamples. We also give a counterexample to regularity in the absence of axi-symmetry. Unlike in the equivalent two dimensional problem, the gradient Holder regularity does not hold for all axi-symmetric solutions, but rather the qualitative regularity properties depend on both the opening angle of the cone and the angle of the oblique vector in the boundary condition.
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We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $mathbb{C}$ that depend smoothly on a real parameter $lambda in [0,1]$. We derive the precise regularity of the solutions in all variables, including the parameter $lambda$. More specifically we show that the solution and its derivatives are continuous in all variables, and the Holder norms of the space variables are bounded uniformly in $lambda$.
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