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A new proof of the boundedness results for stable solutions to semilinear elliptic equations
1. | ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain |
2. | Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain |
3. | BGSMath, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Spain |
We consider the class of stable solutions to semilinear equations $ -\Delta u = f(u) $ in a bounded smooth domain of $ \mathbb{R}^n $. Since 2010 an interior a priori $ L^\infty $ bound for stable solutions is known to hold in dimensions $ n\le 4 $ for all $ C^1 $ nonlinearities $ f $. In the radial case, the same is true for $ n\leq 9 $. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions —for instance $ L^p $ bounds for every finite $ p $ in dimension 5.
Since the mid nineties, the existence of an $ L^\infty $ bound holding for all $ C^1 $ nonlinearities when $ 5\leq n\leq 9 $ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.
References:
[1] |
D. R. Adams,
A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.
doi: 10.1215/S0012-7094-75-04265-9. |
[2] |
H. Brezis, Is there failure of the Inverse Function Theorem?, in Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 1 (2003), 23–33. |
[3] |
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa,
Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.
|
[4] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[5] |
X. Cabré,
Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
doi: 10.1002/cpa.20327. |
[6] |
X. Cabré,
Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.
|
[7] |
X. Cabré and A. Capella,
Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.
doi: 10.1016/j.jfa.2005.12.018. |
[8] |
X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, preprint arXiv: 1907.09403. |
[9] |
X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming. |
[10] |
X. Cabré and G. Poggesi, Stable solutions to some elliptic problems: Minimal cones, the Allen-Cahn equation, and blow-up solutions, Geometry of PDEs and Related Problems, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Cham, 2220 (2018), 1–45. |
[11] |
X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming. |
[12] |
G. Carron,
Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.
|
[13] |
M. G. Crandall and P. H. Rabinowitz,
Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.
doi: 10.1007/BF00280741. |
[14] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143, Boca Raton, FL, 2011.
doi: 10.1201/b10802. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[16] |
P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027. |
[17] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris, 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[18] |
M. Sanchón,
Boundedness of the extremal solution of some $p$-Laplacian problems, Nonlinear Analysis, 67 (2007), 281-294.
doi: 10.1016/j.na.2006.05.010. |
[19] |
P. Sternberg and K. Zumbrun,
Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[20] |
P. Sternberg and K. Zumbrun,
A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.
|
[21] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
doi: 10.1016/j.aim.2012.11.015. |
show all references
References:
[1] |
D. R. Adams,
A note on Riesz potentials, Duke Math. J., 42 (1975), 765-778.
doi: 10.1215/S0012-7094-75-04265-9. |
[2] |
H. Brezis, Is there failure of the Inverse Function Theorem?, in Morse Theory, Minimax Theory and Their Applications to Nonlinear Differential Equations, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 1 (2003), 23–33. |
[3] |
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa,
Blow up for $u_t - \Delta u = g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.
|
[4] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[5] |
X. Cabré,
Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
doi: 10.1002/cpa.20327. |
[6] |
X. Cabré,
Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355-368.
|
[7] |
X. Cabré and A. Capella,
Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.
doi: 10.1016/j.jfa.2005.12.018. |
[8] |
X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, preprint arXiv: 1907.09403. |
[9] |
X. Cabré and P. Miraglio, Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space, forthcoming. |
[10] |
X. Cabré and G. Poggesi, Stable solutions to some elliptic problems: Minimal cones, the Allen-Cahn equation, and blow-up solutions, Geometry of PDEs and Related Problems, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Cham, 2220 (2018), 1–45. |
[11] |
X. Cabré and T. Sanz-Perela, BMO and $L^\infty$ estimates for stable solutions to fractional semilinear elliptic equations, forthcoming. |
[12] |
G. Carron,
Inégalités de Hardy sur les variétés Riemanniennes non-compactes, J. Math. Pures Appl., 76 (1997), 883-891.
|
[13] |
M. G. Crandall and P. H. Rabinowitz,
Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.
doi: 10.1007/BF00280741. |
[14] |
L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143, Boca Raton, FL, 2011.
doi: 10.1201/b10802. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[16] |
P. Miraglio, Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian, preprint arXiv: 1907.13027. |
[17] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris, 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[18] |
M. Sanchón,
Boundedness of the extremal solution of some $p$-Laplacian problems, Nonlinear Analysis, 67 (2007), 281-294.
doi: 10.1016/j.na.2006.05.010. |
[19] |
P. Sternberg and K. Zumbrun,
Connectivity of phase boundaries in strictly convex domains, Arch. Rational Mech. Anal., 141 (1998), 375-400.
doi: 10.1007/s002050050081. |
[20] |
P. Sternberg and K. Zumbrun,
A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85.
|
[21] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
doi: 10.1016/j.aim.2012.11.015. |
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