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The singular set for a semilinear unstable problem. (English) Zbl 1407.35232
In this interesting paper, the author studies the regularity of solutions and the singular points of the following semilinear problem: \[ \Delta u=-\lambda_+(x)(u^+)^q +\lambda_-(x)(u^-)^q \;\;\text{ in } \;\;B_1, \] where \(B_1\) is the unit ball in \(\mathbb{R}^n\), \(0<q<1\) and \(0\leq \lambda_{\pm}\) are Hölder continuous, with \(\lambda_+(x)+\lambda_-(x)\geq\lambda_0>0\) for all \(x\). The main concerns are the local regularity of solutions and their nodal set \(\Gamma=\{u=0\}\).
With the additional assumption that \(u\geq 0\), it was shown in [D. Phillips, Commun. Partial Differ. Equations 8, 1409–1454 (1983; Zbl 0555.35128)] that solutions of \(\Delta u=u^q\) have the optimal \(C^{[\kappa],\kappa-[\kappa]}\) regularity, with \(\kappa=2/(1-q)\). Here, the author studies the singular points at which a solution fails to have this regularity. The singular set is divided in two classes: first, the class containing the points where at least one of the derivatives of order less than \(\kappa\) is nonzero; secondly, the class (denoted by \(\mathcal{S}_{\kappa}\)) of the points where all the derivatives of order less than \(\kappa\) exist and vanish.
The author proves that \(\mathcal{S}_{\kappa}=\emptyset\) when \(\kappa\) is not an integer, and exhibits an example where \(S_{\kappa}\neq\emptyset\) when \(\kappa\in\mathbb{N}\) and \(\lambda_+=\lambda_-\). The crucial step in the proof of the existence of a solution with \(S_{\kappa}\neq\emptyset\) is the classification of \(\kappa\)-homogeneous global solutions in \(\mathbb{R}^2\).
Finally, a regularity investigation when \(n=2\) shows that in this case the singular points in \(\mathcal{S}_{\kappa}\) are isolated. Moreover, if \(x_0\in\mathcal{S}_{\kappa}\), \(\Gamma\setminus\{x_0\}\) is, in a neighborhood of \(x_0\) locally \(C^1\). The main ingredients in this proof are an Almgren type monotonicity formula, along with a Weiss type monotonicity formula.

MSC:
35R35 Free boundary problems for PDEs
35J61 Semilinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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[1] Andersson, J.; Shahgholian, H.; Weiss, SG, Uniform regularity close to cross singularities in an unstable free boundary problem, Commun. Math. Phys., 296, 251-270, (2010) · Zbl 1197.35088
[2] Andersson, J.; Shahgholian, H.; Weiss, SG, On the singularities of a free boundary through fourier expansion, Invent. Math., 187, 535-587, (2012) · Zbl 1234.35318
[3] Andersson, J.; Shahgholian, H.; Weiss, SG, The singular set of higher dimensional unstable obstacle type problems, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 24, 123-146, (2013) · Zbl 1302.35459
[4] Andersson, J.; Weiss, SG, Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem, J. Differ. Equ., 228, 633-640, (2006) · Zbl 1136.35104
[5] Caffarelli, L.; Friedman, A., Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations, J. Differ. Equ., 60, 420-433, (1985) · Zbl 0593.35047
[6] Fotouhi, M.; Shahgolian, H., A semilinear PDE with free boundary, Nonlinear Anal., 151, 145-163, (2017) · Zbl 1357.35138
[7] Monneau, R.; Weiss, GS, An unstable elliptic free boundary problem arising in solid combustion, Duke Math. J., 136, 321-341, (2007) · Zbl 1119.35123
[8] Petrosyan, A., Shahgholian, H., Uraltseva, N.: Regularity of Free Boundaries in Obstacle-Type Problems, Graduate Studies in Mathematics. American Mathematical Society, Providence (2012) · Zbl 1254.35001
[9] Phillips, D., Hausdorff measure estimates of a free boundary for a minimum problem, Commun. Partial Differ. Equ., 8, 1409-1454, (1983) · Zbl 0555.35128
[10] Shahgholian, H., \(C\)1,1 regularity in semilinear elliptic problems, Commun. Pure Appl. Math., 56, 278-281, (2003) · Zbl 1258.35098
[11] Weiss, GS, An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary, Interfaces Free Bound, 3, 121-128, (2001) · Zbl 0986.35139
[12] Weiss, GS, A homogeneity improvement approach to the obstacle problem, Invent. Math., 138, 23-50, (1999) · Zbl 0940.35102
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