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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.
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.
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.
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.
|
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics. Springer-Verlag, (2001).
|
[6] |
M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications., Birkh\, (2006).
|
[7] |
T. Kato, Note on fractional powers of linear operators,, Proc. Japan Acad., 36 (1960), 94.
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, (1968).
|
[9] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, (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.
|
[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., (2001).
|
[12] |
M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations,", World Scientific Publishing Co. Inc., (1998).
|
[13] |
J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265.
|
[14] |
J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 3 (1986), 229.
|
[15] |
J. Moser, A stability theorem for minimal foliations on a torus,, Ergodic Theory Dynam. Systems, (1988), 251.
|
[16] |
J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics,, Springer-Verlag, (1997).
|
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).
|
[18] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (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, (1997).
|
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,", Springer-Verlag, (2001).
|
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).
|
[22] |
M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences., Springer-Verlag, (1996).
|
[23] |
M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences., Springer-Verlag, (1997).
|
[24] |
I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies., North-Holland Publishing Co., (2003).
|
[25] |
K. Yosida, "Functional Analysis,", Springer-Verlag, (1974).
|
show all references
References:
[1] |
S. Bochner, Diffusion equation and stochastic processes,, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368.
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.
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.
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.
|
[5] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Classics in Mathematics. Springer-Verlag, (2001).
|
[6] |
M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications., Birkh\, (2006).
|
[7] |
T. Kato, Note on fractional powers of linear operators,, Proc. Japan Acad., 36 (1960), 94.
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, (1968).
|
[9] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,", Progress in Nonlinear Differential Equations and their Applications, (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.
|
[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., (2001).
|
[12] |
M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations,", World Scientific Publishing Co. Inc., (1998).
|
[13] |
J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265.
|
[14] |
J. Moser, Minimal solutions of variational problems on a torus,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 3 (1986), 229.
|
[15] |
J. Moser, A stability theorem for minimal foliations on a torus,, Ergodic Theory Dynam. Systems, (1988), 251.
|
[16] |
J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics,, Springer-Verlag, (1997).
|
[17] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences,, Springer-Verlag, (1983).
|
[18] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Prentice-Hall Inc., (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, (1997).
|
[20] |
M. A. Shubin, "Pseudodifferential Operators and Spectral Theory,", Springer-Verlag, (2001).
|
[21] |
E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton Mathematical Series, (1970).
|
[22] |
M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences., Springer-Verlag, (1996).
|
[23] |
M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences., Springer-Verlag, (1997).
|
[24] |
I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies., North-Holland Publishing Co., (2003).
|
[25] |
K. Yosida, "Functional Analysis,", Springer-Verlag, (1974).
|
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