Umberto Mosco

Wiener's criterion and Γ-convergence

Dirichlet problems with homogeneous boundary conditions in (possibly irregular) domains and stationary Schrödinger equations with (possibly singular) nonnegative potentials are considered as special cases of more general equations of the form −Δu + µu = 0, whereµ is an arbitrary given nonnegative Borel measure in ℝn. The stability and compactness of weak solutions under suitable variational perturbations ofµ is investigated and stable pointwise estimates for the modulus of continuity and the “energy” of local solutions are obtained.

Sobolev Inequalities on Homogeneous Spaces

We consider a homogeneous spaceX=(X, d, m) of dimension ν≥1 and a local regular Dirichlet forma inL2 (X, m). We prove that if a Poincaré inequality of exponent 1≤p<ν holds on every pseudo-ballB(x, R) ofX, then Sobolev and Nash inequalities of any exponentq∈[p, ν), as well as Poincaré inequalities of any exponentq∈[p, +∞), also hold onB(x, R).

New Directions in Dirichlet Forms

Nonlinear Dirichlet forms by J. Jost From stochastic parallel transport to harmonic maps by W. S. Kendall Dirichlet forms and self-similarity by U. Mosco Stochastic analysis on configuration spaces: Basic ideas and recent results by M. Rockner The geometric aspect of Dirichlet forms by K.-T. Sturm.

A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles

Abstract We prove existence and regularity results for a general inequality including fixed point problems, variational and quasi variational inequalities. Examples of such problems are considered.

Asymptotic Behaviour for Dirichlet Problems in Domains Bounded by Thin Layers

Let Ω ⊂ Rn be a bounded Lipschitz domain surrounded along its boundary by a layer ∑∊ of maximum thickness ∊.

A Derivation Theorem for Capacities with Respect to a Radon Measure

of a Bore1 measure ~1 on R”, n > 2, with respect to a Radon measure v. The measure p is supposed not to charge polar sets but may possibly take the value + cc on subsets of positive capacity. A classical device is provided by the derivation theorems on special families of sets, e.g., the derivation theorems on balls. The density f can be obtained as the limit f(x) = lim inf P(B,(X)) p-0 v(q-4) (1.2)

Composite media and Dirichlet forms

Some relevant “macroscopic” features of bodies with complicated “microscopic” structure are usually described, in the mathematical theory of composite media and homogenization, in terms of asymptotic properties of sequences of Dirichlet integrals

Self-Similar Measures in Quasi-Metric Spaces

{E_h} = \mathop{\smallint }\limits_{\Omega } \sum\limits_{{ij = 1}}^N {{\partial_i}u} {\partial_j}u \,a_h^{{ij}}(x)dx\;,h \in \mathbb{N},