Publicação
Nonlocal Lagrange multipliers and transport densities
| 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. |
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| 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 |
| 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. |
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