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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2015, Volume 27, Issue 3, Pages 311–325 (Mi aa1446)

Research Papers

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

H. Shahgholian

Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden

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.

Keywords: pointwise regularity, Laplace equation, divergence type equations, free boundary problems.

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English version:
St. Petersburg Mathematical Journal, 2016, 27:3, 577–587

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Citation: H. Shahgholian, “Regularity issues for semilinear PDE-s (a narrative approach)”, Algebra i Analiz, 27:3 (2015), 311–325; St. Petersburg Math. J., 27:3 (2016), 577–587

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. E. Indrei, A. Minne, L. Nurbekyan, “Regularity of solutions in semilinear elliptic theory”, Bull. Math. Sci., 7:1 (2017), 177–200
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