Homogenization of Nonlinear PDEs in the Fourier-Stieltjes Algebras

We introduce the Fourier–Stieltjes algebra in $\mathbb{R}^n$ which we denote by $\mathrm{FS}(\mathbb{R}^n)$. It is a subalgebra of the algebra of bounded uniformly continuous functions in $\mathbb{R}^n$, $\mathrm{BUC}(\mathbb{R}^n)$, strictly containing the almost periodic functions, whose elements are invariant by translations and possess a mean value. Thus, it is a so-called algebra with mean value, a concept introduced by Zhikov and Krivenko [Matem. Zametki, 33 (1983), pp. 571–582]. Namely, $\mathrm{FS}(\mathbb{R}^n)$ is the closure in $\mathrm{BUC}(\mathbb{R}^n)$, with the sup norm, of the real-valued functions which may be represented by a Fourier–Stieltjes integral of a complex valued measure with finite total variation. We prove that it is an ergodic algebra and that it shares many interesting properties with the almost periodic functions. In particular, we prove its invariance under the flow of Lipschitz Fourier–Stieltjes fields. We analyze the homogenization problem for nonlinear transport equati...

[1]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  P. Souganidis.  Stochastic homogenization of Hamilton–Jacobi equations and some applications , 1999 .

[4]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[5]  David S. Rosenblum,et al.  Numerical Experiments in Fourier Asymptotics of Cantor Measures and Wavelets , 1992, Exp. Math..

[6]  H. Fédérer.  Geometric Measure Theory , 1969 .

[7]  Anne-Laure Dalibard.  Homogenization of Non-linear Scalar Conservation Laws , 2007 .

[8]  M. Schonbek,et al.  Convergence of solutions to nonlinear dispersive equations , 1982 .

[9]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[10]  L. Ambrosio,et al.  Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications , 2009 .

[11]  R. J. DiPerna.  Convergence of approximate solutions to conservation laws , 1983 .

[12]  R. Strichartz.  Self-similar measures and their Fourier transforms. II , 1993 .

[13]  Joel L. Lebowitz,et al.  Random fields : rigorous results in statistical mechanics and quantum field theory , 1981 .

[14]  E. V. Krivenko,et al.  Averaging of singularly perturbed elliptic operators , 1983 .

[15]  Panagiotis E. Souganidis,et al.  Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media , 2005 .

[16]  I. Gayte,et al.  The two-scale convergence method applied to generalized Besicovitch spaces , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[18]  L. Caffarelli.  Interior a priori estimates for solutions of fully non-linear equations , 1989 .

[19]  H. Ishii.  Almost periodic homogenization of Hamilton-Jacobi equations , 2000 .

[20]  G. Nguetseng,et al.  HOMOGENIZATION OF NONLINEAR MONOTONE OPERATORS BEYOND THE PERIODIC SETTING , 2003 .

[21]  Panagiotis E. Souganidis,et al.  Correctors for the homogenization of Hamilton‐Jacobi equations in the stationary ergodic setting , 2003 .

[22]  L. Caffarelli,et al.  Fully Nonlinear Elliptic Equations , 1995 .

[23]  R. J. Diperna,et al.  Convergence of the viscosity method for isentropic gas dynamics , 1983 .

[24]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[25]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[26]  S. Kozlov,et al.  The method of averaging and walks in inhomogeneous environments , 1985 .

[27]  Silvia Jiménez.  Homogenization of nonlinear partial differential equations , 2010 .

[28]  Alain Bourgeat,et al.  Stochastic two-scale convergence in the mean and applications. , 1994 .

[29]  L. Modica,et al.  Nonlinear Stochastic Homogenization , 1986 .

[30]  J. Carrillo.  Entropy Solutions for Nonlinear Degenerate Problems , 1999 .

[31]  E. Weinan.  Homogenization of linear and nonlinear transport equations , 1992 .

[32]  N. Lloyd,et al.  ALMOST PERIODIC FUNCTIONS AND DIFFERENTIAL EQUATIONS , 1984 .

[33]  A. Besicovitch.  Almost Periodic Functions , 1954 .

[34]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[35]  L. Ambrosio,et al.  Functions of Bounded Variation and Free Discontinuity Problems , 2000 .

[36]  R. J. Diperna,et al.  Measure-valued solutions to conservation laws , 1985 .

[37]  L. Young.  Lectures on the Calculus of Variations and Optimal Control Theory , 1980 .

[38]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[39]  Luc Tartar,et al.  Compensated compactness and applications to partial differential equations , 1979 .

[40]  F. Murat,et al.  Compacité par compensation , 1978 .

[41]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[42]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[43]  L. Ambrosio,et al.  Multiscale Young Measures in almost Periodic Homogenization and Applications , 2009 .

[44]  F. Smithies.  Linear Operators , 2019, Nature.

[45]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .