We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma–Trudinger–Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy (A3w). We use this to give explicit new examples of surplus functions satisfying (A3w), of the form b(x,y) = H(x + y) where H is a convex function on ℝn. We also show that the distribution of equilibrium contracts in this hedonic pricing model is absolutely continuous...

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [Trans. Amer. Math. Soc. 353 (2001) 21–40],...

We prove some optimal regularity results for minimizers of the integral functional $\int f(x,u,Du)\mathrm{d}x$ belonging to the class $K:=\{u\in W^{1,p}\left(\Omega \right)u\ge \psi \}$, where $\psi$ is a fixed function, under standard growth conditions of $p$-type, i.e. $$L^{-1}{\left|z\right|}^p\le f(x,s,z)\le {L(1+|z|}^p).$$

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set...

We consider the problem : (P) Minimize ${\lambda }_2$ over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized...

The aim of this paper is to find the largest lower semicontinuous minorant of the elastic-plastic energy of a body with fissures. The functional of energy considered is not coercive.

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve....

We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type $$𝒥\left(v\right)={\int }_{\Omega }j(x,v,\nabla v)\mathrm{d}x+{\int }_{\Omega }a\left(x\right){\left|v\right|}^2\mathrm{d}x-{\int }_{\Omega }fv\mathrm{d}x,v\in W_0^{1,2}\left(\Omega \right),$$ where $\Omega \subset {ℝ}^N$, $j$ is a Carathéodory function such that $\xi \mapsto j(x,s,\xi )$ is convex, and there exist constants $0\le \tau <1$ and $M>0$ such that $$\frac{{\left|\xi \right|}^2}{{(1+|s\left|\right)}^{\tau }}\le j(x,s,\xi )\le M{\left|\xi \right|}^2$$ for almost all $x\in \Omega$, all $s\in ℝ$ and all $\xi \in {ℝ}^N$. We show that, even if $0<a\left(x\right)$ and $f\left(x\right)$ only belong to $L^1\left(\Omega \right)$, the interplay $$\left|f\right(x\left)\right|\le 2Qa\left(x\right)$$ implies the existence of a minimizer $u\in W_0^{1,2}\left(\Omega \right)$ which belongs to $L^{\infty }\left(\Omega \right)$.