Phương trình Vi phân riêng phần cho Mặt chắn lồi bậc một

Archive for Rational Mechanics and Analysis - Tập 224 - Trang 955-984 - 2017
Adam M. Oberman1, Yuanlong Ruan2
1McGill University Montreal, Canada
2Department of Mathematics, Beihang University, Beijing, China

Tóm tắt

Một phương trình vi phân riêng phần (PDE) cho mặt chắn lồi bậc một được giới thiệu. Sự tồn tại và tính duy nhất của các nghiệm độ bóng cho PDE được thiết lập. Các lược đồ sai phân hữu tỉ elip được xây dựng và sự hội tụ của các nghiệm sai phân hữu tỉ đến nghiệm độ bóng của PDE được chứng minh. Các kết quả tính toán được trình bày và các lớp vật liệu được tính toán từ các mặt chắn. Các kết quả bao gồm ví dụ Kohn–Strang, ví dụ bốn độ dốc cổ điển và một ví dụ với tám độ dốc tạo ra các lớp vật liệu không tầm thường.

Từ khóa

#phương trình vi phân riêng phần #mặt chắn lồi #nghiệm độ bóng #lược đồ sai phân hữu tỉ #hội tụ

Tài liệu tham khảo

Alvarez O., Lasry J.-M., Lions P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. 76(3), 265–288 (1997) Abbasi, B., Oberman, A.M.: A partial differential equation for the strictly quasiconvex envelope. arXiv:1612.06813, 2016 Aranda E., Pedregal P.: Numerical approximation of non-homogeneous, non-convex vector variational problems. Numer. Math. 89(3), 425–444 (2001) Aranda E., Pedregal P.: On the computation of the rank-one convex hull of a function. SIAM J. Sci. Comput. 22(5), 1772–1790 (2001) Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63(4), 337–403, 1976/1977 Bartels S.: Linear convergence in the approximation of rank-one convex envelopes. ESAIM: Math. Model. Numer. Anal. 38(05), 811–820 (2004) Bartels S.: Reliable and efficient approximation of polyconvex envelopes. SIAM J. Numer. Anal. 43(1), 363–385 (2005) Ball, J., James, R.: Fine phase mixtures as minimizers of energy. In: Analysis and Continuum Mechanics, pp. 647–686. Springer, Berlin, 1989 Ball J.M., Kirchheim B., Kristensen J.: Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11(4), 333–359 (2000) Bardi M., Mannucci P.: On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Commun. Pure Appl. Anal. 5(4), 709–731 (2006) Bardi, M., Mannucci, P.: Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of Monge–Ampère type. In: Forum Mathematicum, vol. 25, pp. 1291–1330, 2013 Barles G., Souganidis P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991) Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67, 1992 Chipot M., Kinderlehrer D.: Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103(3), 237–277 (1988) Caffarelli L.A., Nirenberg L., Spruck J.: The dirichlet problem for the degenerate monge-ampère equation. Rev. Mat. Iberoam. 2(1–2), 19–27 (1986) Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, 2nd edn, vol. 78. Springer, Berlin, 2008 Dolzmann, G.: Numerical computation of rank-one convex envelopes. SIAM J. Numer. Anal. 36(5), 1621–1635, 1999 (electronic) Dolzmann, G.: Variational Methods for Crystalline Microstructure-Analysis and Computation. Number 1803. Springer, 2003 De Philippis, G., Figalli, A.: Optimal regularity of the convex envelope. Trans. Am. Math. Soc. 367(6), 4407–4422, 2015 Dolzmann G., Walkington N.J.: Estimates for numerical approximations of rank one convex envelopes. Numer. Math. 85(4), 647–663 (2000) Franěk V., Matoušek J.: Computing d-convex hulls in the plane. Comput. Geom. 42(1), 81–89 (2009) Froese B.D., Oberman A.M.: Convergent filtered schemes for the Monge–Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423–444 (2013) Froese, B.D.: Convergent approximation of surfaces of prescribed Gaussian curvature with weak Dirichlet conditions. arXiv:1601.06315, 2016 Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn, vol. 224. Springer, Berlin, 1983 Kohn R.V., Strang G.: Optimal design and relaxation of variational problems, i. Commun. Pure Appl. Math. 39(1), 113–137 (1986) Kohn R.V., Strang G.: Optimal design and relaxation of variational problems, ii. Commun. Pure Appl. Math. 39(2), 139–182 (1986) Morrey C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2(1), 25–53 (1952) Matoušek J., Plecháč P.: On functional separately convex hulls. Discrete Comput. Geom. 19(1), 105–130 (1998) Muller, S.: Variational models for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Italy, 1996), pp. 85–210. Springer, Berlin, 1999 Motzkin T.S., Wasow W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 31, 253–259 (1953) Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895, 2006 (electronic) Oberman, A.M.: The convex envelope is the solution of a nonlinear obstacle problem. Proc. Am. Math. Soc. 135(6), 1689–1694, 2007 (electronic) Oberman A.M.: Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18(5), 759–780 (2008) Oberman A.M.: Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B, 10(1), 221–238 (2008) Oberman A., Silvestre L.: The Dirichlet problem for the convex envelope. Trans. Am. Math. Soc. 363(11), 5871–5886 (2011) Pedregal, P.: Parametrized Measures and Variational Principles, vol. 30. Springer, 1997