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The obstacle problem for Monge-Ampère type equations in non-convex domains
A comparison principle for a Sobolev gradient semi-flow
1. | Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA Government |
2. | Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States |
3. | Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano, Italy |
We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.
We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.
References:
[1] |
S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368-370.
doi: doi:10.1073/pnas.35.7.368. |
[2] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: doi:10.1002/cpa.20274. |
[3] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.
doi: doi:10.1002/cpa.20093. |
[4] |
E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[6] |
M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006. |
[7] |
T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.
doi: doi:10.3792/pja/1195524082. |
[8] |
O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. |
[9] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. |
[10] |
R. de la Llave and E. Valdinoci, A generalization of aubry-mather theory to partial differential equations and pseudo-differential equations, Annales de l'Institut Henri Poincare C Non Linear Analysis, 26 (2009), 1309-1344. |
[11] |
C. Martínez Carracedo and M. Sanz Alix, "The Theory of Fractional Powers of Operators," volume 187 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 2001. |
[12] |
M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations," World Scientific Publishing Co. Inc., River Edge, NJ, 1998. |
[13] |
J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265-315. |
[14] |
J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272. |
[15] |
J. Moser, A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Systems, $8^*$(Charles Conley Memorial Issue) (1988), 251-281. |
[16] |
J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997. |
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. |
[18] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. |
[19] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," volume 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. |
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer-Verlag, Berlin, second edition, 2001. Translated from the 1978 Russian original by Stig I. Andersson. |
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. |
[22] |
M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996. |
[23] |
M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997. |
[24] |
I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2003. |
[25] |
K. Yosida, "Functional Analysis," Springer-Verlag, New York, fourth edition, 1974. |
show all references
References:
[1] |
S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368-370.
doi: doi:10.1073/pnas.35.7.368. |
[2] |
L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: doi:10.1002/cpa.20274. |
[3] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.
doi: doi:10.1002/cpa.20093. |
[4] |
E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43. |
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[6] |
M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006. |
[7] |
T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96.
doi: doi:10.3792/pja/1195524082. |
[8] |
O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. |
[9] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. |
[10] |
R. de la Llave and E. Valdinoci, A generalization of aubry-mather theory to partial differential equations and pseudo-differential equations, Annales de l'Institut Henri Poincare C Non Linear Analysis, 26 (2009), 1309-1344. |
[11] |
C. Martínez Carracedo and M. Sanz Alix, "The Theory of Fractional Powers of Operators," volume 187 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 2001. |
[12] |
M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations," World Scientific Publishing Co. Inc., River Edge, NJ, 1998. |
[13] |
J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265-315. |
[14] |
J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272. |
[15] |
J. Moser, A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Systems, $8^*$(Charles Conley Memorial Issue) (1988), 251-281. |
[16] |
J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997. |
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. |
[18] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. |
[19] |
R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," volume 49 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. |
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory," Springer-Verlag, Berlin, second edition, 2001. Translated from the 1978 Russian original by Stig I. Andersson. |
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970. |
[22] |
M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996. |
[23] |
M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997. |
[24] |
I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2003. |
[25] |
K. Yosida, "Functional Analysis," Springer-Verlag, New York, fourth edition, 1974. |
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