2000 Mathematics Subject Classification: 35A15, 44A15, 26A33The paper is devoted to the study of the Cauchy-type problem for the differential equation [...] involving the Riemann-Liouville partial fractional derivative of order α > 0 [...] and the Laplace operator.
A system of one-dimensional linear parabolic equations coupled by boundary conditions which include additional state variables, is considered. This system describes an electric circuit with distributed parameter lines and lumped capacitors all connected through a resistive multiport. By using the monotony in a space of the form , one proves the existence and uniqueness of a variational solution, if reasonable engineering hypotheses are fulfilled.
We study the problem of minimizing over the functions that assume given boundary values on . The lagrangian and the domain are assumed convex. A new type of hypothesis on the boundary function is introduced: thelower (or upper) bounded slope condition. This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary...
A corrector theory for the strong approximation of gradient fields inside periodic composites made from two materials with different power law behavior is provided. Each material component has a distinctly different exponent appearing in the constitutive law relating gradient to flux. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied...
Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...
We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.
Using the Maxwell-Higgs model, we prove that linearly unstable symmetric vortices in the Ginzburg-Landau theory are dynamically unstable in the H1 norm (which is the natural norm for the problem).In this work we study the dynamic instability of the radial solutions of the Ginzburg-Landau equations in R2 (...)
We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.