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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 , with the mere assumption that , 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.

**[Ag]**A. Aghajani,*A two-phase free boundary problem for a semilinear elliptic equation*, Bull. Iranian Math. Soc.**40**(2014), no. 5, 1067–1086. MR**3273825****[AP]**H. W. Alt and D. Phillips,*A free boundary problem for semilinear elliptic equations*, J. Reine Angew. Math.**368**(1986), 63–107. MR**850615****[ACF]**Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman,*Variational problems with two phases and their free boundaries*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 431–461. MR**732100**, 10.1090/S0002-9947-1984-0732100-6**[ASW]**John Andersson, Henrik Shahgholian, and Georg S. Weiss,*On the singularities of a free boundary through Fourier expansion*, Invent. Math.**187**(2012), no. 3, 535–587. MR**2891877**, 10.1007/s00222-011-0336-5**[AW1]**J. Andersson and G. S. Weiss,*Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem*, J. Differential Equations**228**(2006), no. 2, 633–640. MR**2289547**, 10.1016/j.jde.2005.11.008**[CP]**L. A. Caffarelli and I. Peral,*On 𝑊^{1,𝑝} estimates for elliptic equations in divergence form*, Comm. Pure Appl. Math.**51**(1998), no. 1, 1–21. MR**1486629**, 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N**[EP]**Anders Edquist and Arshak Petrosyan,*A parabolic almost monotonicity formula*, Math. Ann.**341**(2008), no. 2, 429–454. MR**2385663**, 10.1007/s00208-007-0195-y**[Ev]**Lawrence C. Evans,*A second-order elliptic equation with gradient constraint*, Comm. Partial Differential Equations**4**(1979), no. 5, 555–572. MR**529814**, 10.1080/03605307908820103**[CF]**Luis A. Caffarelli and Avner Friedman,*The free boundary in the Thomas-Fermi atomic model*, J. Differential Equations**32**(1979), no. 3, 335–356. MR**535167**, 10.1016/0022-0396(79)90038-X**[CJK]**Luis A. Caffarelli, David Jerison, and Carlos E. Kenig,*Some new monotonicity theorems with applications to free boundary problems*, Ann. of Math. (2)**155**(2002), no. 2, 369–404. MR**1906591**, 10.2307/3062121**[CSal]**L. Caffarelli and J. Salazar,*Solutions of fully nonlinear elliptic equations with patches of zero gradient: existence, regularity and convexity of level curves*, Trans. Amer. Math. Soc.**354**(2002), no. 8, 3095–3115. MR**1897393**, 10.1090/S0002-9947-02-03009-X**[CSS]**Luis Caffarelli, Jorge Salazar, and Henrik Shahgholian,*Free-boundary regularity for a problem arising in superconductivity*, Arch. Ration. Mech. Anal.**171**(2004), no. 1, 115–128. MR**2029533**, 10.1007/s00205-003-0287-0**[KLS]**S. Kim, K. Lee, and H. Shahgholian,*A free boundary arising from the jump of conductivity*. (to appear)**[Li]**Gary M. Lieberman,*Second order parabolic differential equations*, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR**1465184****[MP]**Norayr Matevosyan and Arshak Petrosyan,*Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients*, Comm. Pure Appl. Math.**64**(2011), no. 2, 271–311. MR**2766528**, 10.1002/cpa.20349**[MP2]**Norayr Matevosyan and Arshak Petrosyan,*Two-phase semilinear free boundary problem with a degenerate phase*, Calc. Var. Partial Differential Equations**41**(2011), no. 3-4, 397–411. MR**2796237**, 10.1007/s00526-010-0367-6**[MW]**R. Monneau and G. S. Weiss,*An unstable elliptic free boundary problem arising in solid combustion*, Duke Math. J.**136**(2007), no. 2, 321–341. MR**2286633**, 10.1215/S0012-7094-07-13624-X**[Na]**Nikolai Nadirashvili,*On stationary solutions of two-dimensional Euler equation*, Arch. Ration. Mech. Anal.**209**(2013), no. 3, 729–745. MR**3067825**, 10.1007/s00205-013-0642-8**[PSU]**Arshak Petrosyan, Henrik Shahgholian, and Nina Uraltseva,*Regularity of free boundaries in obstacle-type problems*, Graduate Studies in Mathematics, vol. 136, American Mathematical Society, Providence, RI, 2012. MR**2962060****[QS]**O. S. de Queiroz and H. Shahgholian,*A free boundary problem with log-term singularity*. (to appear)**[Sh]**Henrik Shahgholian,*𝐶^{1,1} regularity in semilinear elliptic problems*, Comm. Pure Appl. Math.**56**(2003), no. 2, 278–281. MR**1934623**, 10.1002/cpa.10059**[SU]**Henrik Shahgholian and Nina Uraltseva,*Regularity properties of a free boundary near contact points with the fixed boundary*, Duke Math. J.**116**(2003), no. 1, 1–34. MR**1950478**, 10.1215/S0012-7094-03-11611-7**[SUW1]**Henrik Shahgholian, Nina Uraltseva, and Georg S. Weiss,*Global solutions of an obstacle-problem-like equation with two phases*, Monatsh. Math.**142**(2004), no. 1-2, 27–34. MR**2065019**, 10.1007/s00605-004-0235-6**[SUW2]**Henrik Shahgholian, Nina Uraltseva, and Georg S. Weiss,*The two-phase membrane problem—regularity of the free boundaries in higher dimensions*, Int. Math. Res. Not. IMRN**8**(2007), Art. ID rnm026, 16. MR**2340105**, 10.1093/imrn/rnm026**[SUW3]**Henrik Shahgholian, Nina Uraltseva, and Georg S. Weiss,*A parabolic two-phase obstacle-like equation*, Adv. Math.**221**(2009), no. 3, 861–881. MR**2511041**, 10.1016/j.aim.2009.01.011**[TZ]**Eduardo V. Teixeira and Lei Zhang,*A local parabolic monotonicity formula on Riemannian manifolds*, J. Geom. Anal.**21**(2011), no. 3, 513–526. MR**2810841**, 10.1007/s12220-010-9156-x**[Ur]**N. N. Uraltseva,*Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities*, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 235–246. MR**2343613****[Ye]**Karen Yeressian,*Nondegeneracy in the obstacle problem with a degenerate force term*, Interfaces Free Bound.**17**(2015), no. 2, 233–244. MR**3391970**, 10.4171/IFB/340

Retrieve articles in *St. Petersburg Mathematical Journal*
with MSC (2010):
35J61,
35K58

Retrieve articles in all journals with MSC (2010): 35J61, 35K58

Additional Information

**H. Shahgholian**

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

Email:
henriksh@kth.se

DOI:
https://doi.org/10.1090/spmj/1405

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