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)
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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:https://bibliotecadigital.exactas.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