Publicação

Nonlocal Lagrange multipliers and transport densities

Ver documento

Detalhes bibliográficos
Resumo:We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional \( s \)-gradient constraint, \( 0 < s < 1 \), associated to a general, possibly degenerate, linear fractional operator of the type, $$ L_s u = -D^s \cdot (A D^s u + bu) + d \cdot D^s u + cu, $$ with integrable data, in the space \( \Lambda^{s,p}_0(\Omega) \), which is the completion of the set of smooth functions with compact support in a bounded domain \( \Omega \) for the \( L_p \)-norm of the distributional Riesz fractional gradient \( D^s \) in \( \mathbb{R}^d \) (when \( s = 1 \), \( D_1 = D \) is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of \( L^\infty(\mathbb{R}^d) \) and are associated to the variational inequalities of the corresponding transport potentials under the constraint \( |D^s u| \leq g \). Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator \( L_s u \). For this purpose, we also develop some relevant properties of the spaces \( \Lambda^{s,p}_0(\Omega) \), including the limit case \( p = \infty \) and the continuous embeddings \( \Lambda^{s,q}_0(\Omega) \subset \Lambda^{s,p}_0(\Omega) \), for \( 1 \leq p \leq q \leq \infty \). We also show the localisation of the nonlocal problems \( (0 < s < 1) \), to the local limit problem with classical gradient constraint when \( s \rightarrow 1 \), for which most results are also new for a general, possibly degenerate, partial differential operator \( L_1 u \) only with integrable coeficients and bounded gradient constraint.
Autores principais:Azevedo, Assis
Outros Autores:Rodrigues, José Francisco; Santos, Lisa
Assunto:Fractional gradient Nonlocal variational inequalities Lagrange multipliers
Ano:2024
País:Portugal
Tipo de documento:artigo
Tipo de acesso:acesso aberto
Instituição associada:Universidade do Minho
Idioma:inglês
Origem:RepositóriUM - Universidade do Minho

Registos relacionados

Registos relacionados