Minimal strictly stable solutions of -Δu=f(u) in smooth uniformly convex planar domains can have nonconvex superlevel sets for f(u)=e^u or f(u)=(a+u)^p, answering Brezis's open question negatively.
Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
This paper focuses on establishing the existence of a class of steady solutions, termed least total curvature solutions, to the incompressible Euler system in a strip. The solutions obtained in this paper complement the least total curvature solutions already known. Our approach employs a minimization procedure to identify a monotone heteroclinic solution for a conveniently chosen semilinear elliptic PDE. This method also enables us to construct positive and monotone (and consequently stable) solutions to semilinear elliptic PDEs with non-convex superlevel sets in a strip domain. This can be regarded as a negative answer to a generalized problem raised in [27].
fields
math.AP 2years
2026 2representative citing papers
Vanishing viscosity selects constant-vorticity flows in bounded domains and shear flows in strips as the only possible limits for 2D steady Euler equations.
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Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
Minimal strictly stable solutions of -Δu=f(u) in smooth uniformly convex planar domains can have nonconvex superlevel sets for f(u)=e^u or f(u)=(a+u)^p, answering Brezis's open question negatively.
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A selection principle for 2D steady Euler flows via the vanishing viscosity limit
Vanishing viscosity selects constant-vorticity flows in bounded domains and shear flows in strips as the only possible limits for 2D steady Euler equations.