Fractional problems in thin domains

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In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂R n and V⊂R m , we show that the solution u ε to Δ x s u ε (x,y)+Δ y t u ε (x,y)=f(x,ε −1 y)inU×εV with u ε (x,y)=0 if x⁄∈U and y∈εV, verifies that ũ ε (x,y)≔u ε (x,εy)→u 0 strongly in the natural fr...

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Autor principal: Pereira, M.C
Otros Autores: Rossi, J.D, Saintier, N.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Elsevier Ltd 2019
Acceso en línea:Registro en Scopus
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100 1 |a Pereira, M.C. 
245 1 0 |a Fractional problems in thin domains 
260 |b Elsevier Ltd  |c 2019 
270 1 0 |m Rossi, J.D.; Dpto. de Matemáticas, FCEyN, Universidad de Buenos AiresArgentina; email: jrossi@dm.uba.ar 
506 |2 openaire  |e Política editorial 
504 |a Shuichi, J., Morita, Y., Remarks on the behavior of certain eigenvalues on a singularly perturbed domain with several thin channels (1991) Comm. Part. Diff. Eq., 17 (3), pp. 189-226 
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504 |a Prizzi, M., Rybakowski, K.P., Recent results on thin domain problems ii (2002) Top. Meth. Nonlinear Anal., 19, pp. 199-219 
504 |a Ferreira, R., Mascarenhas, M.L., Piatnitski, A., Spectral analysis in thin tubes with axial heterogeneities (2015) Portugal. Math., 72, pp. 247-266 
504 |a Pereira, M.C., Silva, R.P., Remarks on the p-Laplacian on thin domains (2015) Progr. Nonlinear Differential Equations Appl., pp. 389-403 
504 |a Barros, S.R.M., Pereira, M.C., Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary (2016) J. Math. Anal. Appl., 441 (1), pp. 375-392 
504 |a Arrieta, J.M., Villanueva-Pesqueira, M., Thin domains with non-smooth periodic oscillatory boundaries (2017) J. Math. Anal. Appl., 446, pp. 130-164 
504 |a Saintier, N., Asymptotics of best Sobolev constants on thin manifolds (2009) J. Differential Equations, 246, pp. 2876-2890 
504 |a Aris, R., On the dispersion of a solute in a fluid flowing through a tube (1956) Proc. Roy. Soc. London Sect. A, 235, pp. 67-77 
504 |a Iftimie, D., The 3D Navier–Stokes equations seen as a perturbation of the 2D Navier- Stokes equations (1999) Bull. Soc. Math. France, 127, pp. 473-518 
504 |a Hong, L.T., Sell, G.R., Navier–stokes equations with navier boundary conditions for an oceanic model (2010) J. Dynam. Differential Equations, 22 (3), pp. 563-616 
504 |a Bella, P., Feireisl, E., Novotny, A., Dimension reduction for compressible viscous fluids (2014) Acta Appl. Math., 134, pp. 111-121 
504 |a Fabricius, J., Koroleva, Y.O., Tsandzana, A., Wall, P., Asymptotic behavior of Stokes flow in a thin domain with a moving rough boundary (2014) Proc. R. Soc. A, 470 
504 |a Benes, M., Pazanin, I., Suárez-Grau, F.J., Heat flow through a thin cooled pipe filled with a micropolar fluid (2015) J. Theoret. Appl. Mech., 53 (3), pp. 569-579 
504 |a Liao, X., On the strong solutions of the inhomogeneous incompressible Navier–Stokes equations in a thin domain (2016) Differential Integral Equations, 29, pp. 167-182 
504 |a Pereira, M.C., Rossi, J.D., Nonlocal problems in thin domains. Preprint; DiNezza, E., Palatucci, G., Valdinoci, E., Hitchhiker's guide to the fractional Sobolev spaces (2012) Bull. Sci. Math., 136, pp. 521-573 
504 |a Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo, J., (2010) Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. , AMS 
504 |a Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N., How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems (2007) Arch. Ration. Mech. Anal., 187, pp. 137-156 
504 |a Cortazar, C., Elgueta, M., Rossi, J.D., Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions (2009) Israel J. Math., 170 (1), pp. 53-60 
504 |a Pereira, M.C., Silva, R.P., Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure. Quart (2015) Appl. Math., 73, pp. 537-552 
504 |a Raugel, G., (1995) Dynamics of Partial Differential Equations on Thin Domains, Lecture Notes in Mathematics, 1609. , Springer-Verlag 
504 |a Ladyzhenskaya, O.A., Ural'tseva, N.N., Linear and quasilinear elliptic equations. Vol. 46 (1968), Academic Press; MolicaBisci, G., Radulescu, V., Servadei, R., Variational methods for nonlocal fractional problems. With a foreword by Jean Mawhin (2016) Encyclopedia of Mathematics and its Applications, Vol. 162, p. xvi+383. , Cambridge University Press Cambridge 
520 3 |a In this paper we consider nonlocal fractional problems in thin domains. Given open bounded subsets U⊂R n and V⊂R m , we show that the solution u ε to Δ x s u ε (x,y)+Δ y t u ε (x,y)=f(x,ε −1 y)inU×εV with u ε (x,y)=0 if x⁄∈U and y∈εV, verifies that ũ ε (x,y)≔u ε (x,εy)→u 0 strongly in the natural fractional Sobolev space associated to this problem. We also identify the limit problem that is satisfied by u 0 and estimate the rate of convergence in the uniform norm. Here Δ x s u and Δ y t u are the fractional Laplacian in the 1st variable x (with a Dirichlet condition, u(x)=0 if x⁄∈U) and in the 2nd variable y (with a Neumann condition, integrating only inside V), respectively, that is, Δ x s u(x,y)=∫ R n [Formula presented]dw and Δ y t u(x,y)=∫ V [Formula presented]dz. © 2019 Elsevier Ltd  |l eng 
536 |a Detalles de la financiación: Conselho Nacional de Desenvolvimento Científico e Tecnológico, 302960/2014-7 
536 |a Detalles de la financiación: Fundação de Amparo à Pesquisa do Estado de São Paulo, 2015/17702-3 
536 |a Detalles de la financiación: Ministerio de Ciencia, Tecnología e Innovación Productiva, MTM2016-68210 
536 |a Detalles de la financiación: The first author (MCP) is partially supported by CNPq (Brazil) 302960/2014-7 and FAPESP 2015/17702-3 (Brazil) and the second author (JDR) by MINCYT (Argentina) grant MTM2016-68210 (Spain). 
593 |a Dpto. de Matemática Aplicada, IME, Universidade de São Paulo, Rua do Matão, São Paulo - SP 1010, Brazil 
593 |a Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina 
690 1 0 |a DIRICHLET PROBLEM 
690 1 0 |a NEUMANN PROBLEM 
690 1 0 |a NONLOCAL FRACTIONAL EQUATIONS 
690 1 0 |a THIN DOMAINS 
690 1 0 |a BOUNDARY VALUE PROBLEMS 
690 1 0 |a DIRICHLET CONDITION 
690 1 0 |a DIRICHLET PROBLEM 
690 1 0 |a FRACTIONAL EQUATION 
690 1 0 |a FRACTIONAL LAPLACIAN 
690 1 0 |a NEUMANN PROBLEM 
690 1 0 |a OPEN BOUNDED SUBSETS 
690 1 0 |a RATE OF CONVERGENCE 
690 1 0 |a THIN DOMAINS 
690 1 0 |a SOBOLEV SPACES 
700 1 |a Rossi, J.D. 
700 1 |a Saintier, N. 
773 0 |d Elsevier Ltd, 2019  |p Nonlinear Anal Theory Methods Appl  |x 0362546X  |w (AR-BaUEN)CENRE-254  |t Nonlinear Analysis, Theory, Methods and Applications 
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