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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

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 & 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

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M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006.  Google Scholar

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T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96. doi: doi:10.3792/pja/1195524082.  Google Scholar

[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.  Google Scholar

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A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations," World Scientific Publishing Co. Inc., River Edge, NJ, 1998.  Google Scholar

[13]

J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265-315.  Google Scholar

[14]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  Google Scholar

[15]

J. Moser, A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Systems, $8^*$(Charles Conley Memorial Issue) (1988), 251-281.  Google Scholar

[16]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar

[18]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.  Google Scholar

[23]

M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997.  Google Scholar

[24]

I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[25]

K. Yosida, "Functional Analysis," Springer-Verlag, New York, fourth edition, 1974.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[6]

M. Haase, "The Functional Calculus for Sectorial Operators," volume 169 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 2006.  Google Scholar

[7]

T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad., 36 (1960), 94-96. doi: doi:10.3792/pja/1195524082.  Google Scholar

[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.  Google Scholar

[9]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Miklavčič, "Applied Functional Analysis and Partial Differential Equations," World Scientific Publishing Co. Inc., River Edge, NJ, 1998.  Google Scholar

[13]

J. Moser, A rapidly convergent iteration method and non-linear partial differential equations. I, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 265-315.  Google Scholar

[14]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 229-272.  Google Scholar

[15]

J. Moser, A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Systems, $8^*$(Charles Conley Memorial Issue) (1988), 251-281.  Google Scholar

[16]

J. W. Neuberger, "Sobolev Gradients and Differential Equations," volume 1670 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997.  Google Scholar

[17]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.  Google Scholar

[18]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Prentice-Hall Inc., Englewood Cliffs, N.J., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

M. E. Taylor, "Partial differential equations. I," volume 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996.  Google Scholar

[23]

M. E. Taylor, "Partial Differential Equations. III," volume 117 of Applied Mathematical Sciences. Springer-Verlag, New York, 1997.  Google Scholar

[24]

I. I. Vrabie, "$C_0$-semigroups and Applications," volume 191 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[25]

K. Yosida, "Functional Analysis," Springer-Verlag, New York, fourth edition, 1974.  Google Scholar

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