Communications in Partial Differential Equations, 1994
ABSTRACT The author considers the evolution problem ∂u ∂t-Δu=|Du| 2 u,u(x,0)=u 0 (x)(1) for mappi... more ABSTRACT The author considers the evolution problem ∂u ∂t-Δu=|Du| 2 u,u(x,0)=u 0 (x)(1) for mappings u:Ω×(0,∞)→S m-1 where Ω denotes an open subset of ℝ n and S m-1 is the unit sphere. Y. Chen and M. Struwe [Math. Z. 201, No. 1, 83-103 (1989; Zbl 0666.58016)] produced a weak solution of (1) with certain partial regularity properties, J.-M. Coron [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 4, 335-344 (1990; Zbl 0707.58017)] showed that problem (1) has infinitely many solutions different from the one constructed by Chen and Struwe, and in general no partial regularity theory is available. In this paper the author imposes an additional condition on the class of weak solutions which he calls stability hypothesis and which holds true for smooth solutions. Stability should be seen as an analog to stationarity for harmonic maps and in fact stability is sufficient for partial regularity for solutions of (1).
ABSTRACT This paper concerns the two-phase problem F(D 2 u,Du)=0 in Ω + ∪Ω - , u ν + =G(u ν - ) o... more ABSTRACT This paper concerns the two-phase problem F(D 2 u,Du)=0 in Ω + ∪Ω - , u ν + =G(u ν - ) on Γ=∂Ω + ∖∂Ω, u=φ on ∂Ω, where Ω is a domain in ℝ n with smooth boundary, Ω + =Ω∩{u>0}, Ω - =Ω∖Ω + ¯, and the subscript ν stands for the normal derivative at the free boundary Γ. By assuming that F(·,·) satisfies certain elliptic conditions (possibly nonconcavity) and G(·) is increasing, the author proves that the Lipschitz free boundary is in fact C 1,α with some α>0.
Communications in Partial Differential Equations, 1994
ABSTRACT The author considers the evolution problem ∂u ∂t-Δu=|Du| 2 u,u(x,0)=u 0 (x)(1) for mappi... more ABSTRACT The author considers the evolution problem ∂u ∂t-Δu=|Du| 2 u,u(x,0)=u 0 (x)(1) for mappings u:Ω×(0,∞)→S m-1 where Ω denotes an open subset of ℝ n and S m-1 is the unit sphere. Y. Chen and M. Struwe [Math. Z. 201, No. 1, 83-103 (1989; Zbl 0666.58016)] produced a weak solution of (1) with certain partial regularity properties, J.-M. Coron [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 4, 335-344 (1990; Zbl 0707.58017)] showed that problem (1) has infinitely many solutions different from the one constructed by Chen and Struwe, and in general no partial regularity theory is available. In this paper the author imposes an additional condition on the class of weak solutions which he calls stability hypothesis and which holds true for smooth solutions. Stability should be seen as an analog to stationarity for harmonic maps and in fact stability is sufficient for partial regularity for solutions of (1).
ABSTRACT This paper concerns the two-phase problem F(D 2 u,Du)=0 in Ω + ∪Ω - , u ν + =G(u ν - ) o... more ABSTRACT This paper concerns the two-phase problem F(D 2 u,Du)=0 in Ω + ∪Ω - , u ν + =G(u ν - ) on Γ=∂Ω + ∖∂Ω, u=φ on ∂Ω, where Ω is a domain in ℝ n with smooth boundary, Ω + =Ω∩{u>0}, Ω - =Ω∖Ω + ¯, and the subscript ν stands for the normal derivative at the free boundary Γ. By assuming that F(·,·) satisfies certain elliptic conditions (possibly nonconcavity) and G(·) is increasing, the author proves that the Lipschitz free boundary is in fact C 1,α with some α>0.
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Papers by Mikhail Feldman