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1.
We give, in a non-smooth setting, some conditions under which (some of) the minimizers of among the functions in W1,1(Ω) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function are bounded and the boundary datum u0 satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of on , whenever they exist, are Lipschitz. A relaxation result follows. 相似文献
2.
Given a strictly convex, smooth, and bounded domain Ω in we establish the existence of a negative convex solution in with zero boundary value to the singular Monge–Ampère equation det(D2u)=p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates. 相似文献
3.
Thierry Cazenave Flvio Dickstein Fred B. Weissler 《Journal of Mathematical Analysis and Applications》2009,360(2):537-547
In this paper, we consider the nonlinear heat equation(NLH)
ut−Δu=|u|αu,