A comparison principle for a Sobolev gradient semi-flow

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January 2011, 10(1): 69-91. doi: 10.3934/cpaa.2011.10.69

A comparison principle for a Sobolev gradient semi-flow

Timothy Blass ^1, , Rafael De La Llave ^2, and Enrico Valdinoci ^3,

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

Received January 2010 Revised June 2010 Published November 2010

Abstract
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We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.
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.

Keywords: Comparison principle, fractional powers of elliptic operators., Sobolev gradient, semigroups of linear operators.

Mathematics Subject Classification: Primary: 35B50, 46N20; Secondary: 35J2.

Citation: Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure and Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69

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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