Utilize este identificador para referenciar este registo:
https://hdl.handle.net/1822/88311
Título: | Nonlocal Lagrange multipliers and transport densities |
Autor(es): | Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
Palavras-chave: | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
Data: | 2023 |
Editora: | World Scientific |
Revista: | Bulletin of Mathematical Sciences |
Citação: | Azevedo, A., Rodrigues, J. F., & Santos, L. (2023, December 8). Nonlocal Lagrange multipliers and transport densities. Bulletin of Mathematical Sciences. World Scientific Pub Co Pte Ltd. http://doi.org/10.1142/s1664360723500145 |
Resumo(s): | 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, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs bu)+\bs d\cdot D^su+cu , \end{equation*} 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 $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. 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 $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint. |
Tipo: | Artigo |
URI: | https://hdl.handle.net/1822/88311 |
DOI: | 10.1142/S1664360723500145 |
ISSN: | 1664-3607 |
e-ISSN: | 1664-3615 |
Versão da editora: | https://www.worldscientific.com/doi/10.1142/S1664360723500145 |
Arbitragem científica: | yes |
Acesso: | Acesso aberto |
Aparece nas coleções: | CMAT - Artigos em revistas com arbitragem / Papers in peer review journals |
Ficheiros deste registo:
Ficheiro | Descrição | Tamanho | Formato | |
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Azevedo_Rodrigues_Santos_Final.pdf | 617,18 kB | Adobe PDF | Ver/Abrir |