Artículo

Barrios, B.; Del Pezzo, L.; García-Melián, J.; Quaas, A. "Monotonicity of solutions for some nonlocal elliptic problems in half-spaces" (2017) Calculus of Variations and Partial Differential Equations. 56(2)
Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

In this paper we consider classical solutions u of the semilinear fractional problem (- Δ) su= f(u) in R+N with u= 0 in RN\\R+N, where (- Δ) s, 0 < s< 1 , stands for the fractional laplacian, N≥ 2 , R+N={x=(x′,xN)∈RN:xN>0} is the half-space and f∈ C1 is a given function. With no additional restriction on the function f, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in R+N and verify (Formula presented.). This is in contrast with previously known results for the local case s= 1 , where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when f(0) < 0. © 2017, Springer-Verlag Berlin Heidelberg.

Registro:

Documento: Artículo
Título:Monotonicity of solutions for some nonlocal elliptic problems in half-spaces
Autor:Barrios, B.; Del Pezzo, L.; García-Melián, J.; Quaas, A.
Filiación:Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, La Laguna, 38200, Spain
Departamento de Matemática FCEyN, UBA, CONICET, Ciudad Universitaria Pab I, Buenos Aires, 1428, Argentina
Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, La Laguna, 38200, Spain
Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680, Valparaiso, Chile
Palabras clave:35S15; 45M20; 47G10
Año:2017
Volumen:56
Número:2
DOI: http://dx.doi.org/10.1007/s00526-017-1133-9
Título revista:Calculus of Variations and Partial Differential Equations
Título revista abreviado:Calc. Var. Partial Differ. Equ.
ISSN:09442669
Registro:http://digital.bl.fcen.uba.ar/collection/paper/document/paper_09442669_v56_n2_p_Barrios

Referencias:

  • Alberti, G., Bellettini, G., A nonlocal anisotropic model for phase transitions. I. The optimal profile problem (1998) Math. Ann., 310 (3), pp. 527-560
  • Applebaum, D., (2009) Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, , 2, Cambridge University Press, Cambridge
  • Berestycki, H., Caffarelli, L., Nirenberg, L., Symmetry for elliptic equations in a half-space (1993) Boundary Value Problems for Partial Differential Equations and Applications, Research Notes in Applied Mathematics, pp. 27-42. , Lions JL, (ed), 29, Masson, Paris
  • Berestycki, H., Caffarelli, L., Nirenberg, L., Inequalities for second-order elliptic equations with applications to unbounded domains I (1996) Duke Math. J., 81, pp. 467-494
  • Berestycki, H., Caffarelli, L., Nirenberg, L., Further qualitative properties for elliptic equations in unbounded domains (1997) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25, pp. 69-94
  • Berestycki, H., Caffarelli, L., Nirenberg, L., Monotonicity for elliptic equations in unbounded Lipschitz domains (1997) Comm. Pure Appl. Math., 50, pp. 1089-1111
  • Bertoin, J., (1996) Lévy Processes”, Cambridge Tracts in Mathematics, 121, , Cambridge University Press, Cambridge
  • Blumenthal, R.M., Getoor, R.K., Ray, D.B., On the distribution of the first hits for the symmetric stable processes (1961) Trans. Am. Math. Soc., 99, pp. 540-554
  • Bouchaud, J.P., Georges, A., Anomalous diffusion in disordered media, statistical mechanics, models and physical applications (1990) Phys. Rep., 195, p. 127
  • Cabré, X., Sola-Morales, J., Layer solutions in a half-space for boundary reactions (2005) Comm. Pure Appl. Math., 58 (12), pp. 1678-1732
  • Caffarelli, L., Further regularity for the Signorini problem (1979) Comm. Partial Differ. Equ., 4 (9), pp. 1067-1075
  • Caffarelli, L., Roquejoffre, J.M., Sire, Y., Variational problems with free boundaries for the fractional Laplacian (2010) J. Eur. Math. Soc., 12 (5), pp. 1151-1179
  • Caffarelli, L., Silvestre, L., Regularity theory for fully nonlinear integro differential equations (2009) Comm. Pure Appl. Math., 62 (5), pp. 597-638
  • Caffarelli, L., Silvestre, L., Regularity results for nonlocal equations by approximation (2011) Arch. Ration. Mech. Anal., 200, pp. 59-88
  • Caffarelli, L., Vasseur, L., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation (2010) Ann. Math. (2), 171 (3), pp. 1903-1930
  • Chen, W., Fang, Y., Yang, R., Liouville theorems involving the fractional Laplacian on a half space (2015) Adv. Math., 274, pp. 167-198
  • Constantin, P., Euler equations, Navier-Stokes equations and turbulence (2006) Mathematical foundation of turbulent viscous flows”, Vol. 1871 of Lecture Notes in Mathematics, , Springer, Berlin
  • Cont, R., Tankov, P., (2004) Financial Modelling with Jump Processes, CRC Financial Mathematics Series, , Chapman & Hall, Boca Raton
  • Cortázar, C., Elgueta, M., García-Melián, J., Nonnegative solutions of semilinear elliptic equations in half-spaces (2016) J. Math. Pures Appl., 106, pp. 866-876
  • Dancer, N., On the number of positive solutions of weakly non-linear elliptic equations when a parameter is large (1986) Proc. Lond. Math. Soc., 53, pp. 429-452
  • Dancer, N., Some notes on the method of moving planes (1992) Bull. Aust. Math. Soc., 46 (3), pp. 425-434
  • Dipierro, S., Figalli, A., Valdinoci, E., Strongly nonlocal dislocation dynamics in crystals (2014) Comm. Partial Differ. Equ., 39 (12), pp. 2351-2387
  • Dipierro, S., Soave, N., Valdinoci, E., On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math (2016) Ann
  • Dupaigne, L., Sire, Y., A Liouville theorem for non local elliptic equations (2010) Symmetry for Elliptic PDEs, Contemporary Mathematics, , Farina A, Valdinoci E, (eds), 528, American Mathematical Society, Providence
  • Fall, M.M., Weth, T., Monotonicity and nonexistence results for some fractional elliptic problems in the half space (2016) Comm. Contemp. Math., 18, p. 1550012. , (25 pages)
  • Farina, A., Sciunzi, B., Qualitative properties and classification of nonnegative solutions to - Δ u= f(u) in unbounded domains when f(0) < 0 (2016) Rev. Mat. Iberoam., , 32(4), 1311–1330
  • Farina, A., Soave, N., Symmetry and uniqueness of nonnegative solutions of some problems in the halfspace (2013) J. Math. Anal. Appl., 403, pp. 215-233
  • Felmer, P., Wang, Y., Radial symmetry of positive solutions to equations involving the fractional laplacian (2014) Comm. Contemp. Math., 16, p. 1350023. , (24 pages)
  • Principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term., , http://arxiv.org/abs/1605.09787, Quaas, A., Salort, A., Xia, A
  • Quaas, A., Xia, A., Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space (2015) Calc. Var. Part. Diff. Equ., 52, pp. 641-659
  • Ros-Oton, X., Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary (2014) J. Math. Pures Appl., 101, pp. 275-302
  • Savin, O., Valdinoci, E., Elliptic PDEs with fibered nonlinearities (2009) J. Geom. Anal., 19 (2), pp. 420-432
  • Servadei, R., Valdinoci, E., Variational methods for non-local operators of elliptic type (2013) Discret. Cont. Dyn. Syst., 33, pp. 2105-2137
  • Signorini, A., Questioni di elasticitá non linearizzata e semilinearizzata (1959) Rendiconti di Matematica e delle sue applicazioni, 18, pp. 95-139
  • Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator (2007) Comm. Pure Appl. Math., 60 (1), pp. 67-112
  • Sire, Y., Valdinoci, E., Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result (2009) J. Funct. Anal., 256 (6), pp. 1842-1864
  • Tarasov, V., Zaslasvky, G., Fractional dynamics of systems with long-range interaction (2006) Comm. Nonl. Sci. Numer. Simul., 11, pp. 885-889
  • Toland, J., The Peierls–Nabarro and Benjamin–Ono equations (1997) J. Funct. Anal., 145 (1), pp. 136-150

Citas:

---------- APA ----------
Barrios, B., Del Pezzo, L., García-Melián, J. & Quaas, A. (2017) . Monotonicity of solutions for some nonlocal elliptic problems in half-spaces. Calculus of Variations and Partial Differential Equations, 56(2).
http://dx.doi.org/10.1007/s00526-017-1133-9
---------- CHICAGO ----------
Barrios, B., Del Pezzo, L., García-Melián, J., Quaas, A. "Monotonicity of solutions for some nonlocal elliptic problems in half-spaces" . Calculus of Variations and Partial Differential Equations 56, no. 2 (2017).
http://dx.doi.org/10.1007/s00526-017-1133-9
---------- MLA ----------
Barrios, B., Del Pezzo, L., García-Melián, J., Quaas, A. "Monotonicity of solutions for some nonlocal elliptic problems in half-spaces" . Calculus of Variations and Partial Differential Equations, vol. 56, no. 2, 2017.
http://dx.doi.org/10.1007/s00526-017-1133-9
---------- VANCOUVER ----------
Barrios, B., Del Pezzo, L., García-Melián, J., Quaas, A. Monotonicity of solutions for some nonlocal elliptic problems in half-spaces. Calc. Var. Partial Differ. Equ. 2017;56(2).
http://dx.doi.org/10.1007/s00526-017-1133-9