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Regularity issues for semilinear PDE-s (a narrative approach)

Author: H. Shahgholian
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 577-587
MSC (2010): Primary 35J61, 35K58
Published electronically: March 30, 2016
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Abstract: Occasionally, solutions of semilinear equations have better (local) regularity properties than the linear ones if the equation is independent of space (and time) variables. The simplest example, treated by the current author, was that the solutions of $ \Delta u = f(u)$, with the mere assumption that $ f'\geq -C$, have bounded second derivatives. In this paper, some aspects of semilinear problems are discussed, with the hope to provoke a study of this type of problems from an optimal regularity point of view. It is noteworthy that the above result has so far been undisclosed for linear second order operators, with Hölder coefficients. Also, the regularity of level sets of solutions as well as related quasilinear problems are discussed. Several seemingly plausible open problems that might be worthwhile are proposed.

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Additional Information

H. Shahgholian
Affiliation: Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

Keywords: Pointwise regularity, Laplace equation, divergence type equations, free boundary problems
Received by editor(s): March 2, 2015
Published electronically: March 30, 2016
Additional Notes: Supported in part by Swedish Research Council
Dedicated: Dedicated to Nina Nikolaevna Ural’tseva
Article copyright: © Copyright 2016 American Mathematical Society