Article
| Title: | A note on the shift theorem for the Laplacian in polygonal domains (English) |
| Author: | Melenk, Jens Markus |
| Author: | Rojik, Claudio |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 69 |
| Issue: | 5 |
| Year: | 2024 |
| Pages: | 653-693 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces. (English) |
| Keyword: | Besov space |
| Keyword: | corner domain |
| Keyword: | corner singularity |
| Keyword: | Mellin calculus |
| MSC: | 35B65 |
| MSC: | 35J25 |
| DOI: | 10.21136/AM.2024.0049-24 |
| . | |
| Date available: | 2024-11-05T12:04:48Z |
| Last updated: | 2024-11-05 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152636 |
| . | |
| Reference: | [1] Babuška, I., Guo, B. Q.: Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order.SIAM J. Math. Anal. 19 (1988), 172-203. Zbl 0647.35021, MR 0924554, 10.1137/0519014 |
| Reference: | [2] Babuška, I., Guo, B. Q.: Regularity of the solution of elliptic problems with piecewise analytic data. II. The trace spaces and application to the boundary value problems with nonhomogeneous boundary conditions.SIAM J. Math. Anal. 20 (1989), 763-781. Zbl 0706.35028, MR 1000721, 10.1137/0520054 |
| Reference: | [3] Babuška, I., Kellogg, R. B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements.Numer. Math. 33 (1979), 447-471. Zbl 0423.65057, MR 0553353, 10.1007/BF01399326 |
| Reference: | [4] Babuška, I., Osborn, J.: Eigenvalue problems.Finite Element Methods 1 Handbook of Numerical Analysis II. North Holland, Amsterdam (1991), 641-789. Zbl 0875.65087, MR 1115240 |
| Reference: | [5] Bacuta, C.: Interpolation Between Subspaces of Hilbert Spaces and Applications to Shift Theorems for Elliptic Boundary Value Problems and Finite Element Methods: Ph. D. Thesis.Texas A&M University, College Station (2000). MR 2701566 |
| Reference: | [6] Bacuta, C., Bramble, J. H., Xu, J.: Regularity estimates for elliptic boundary value problems in Besov spaces.Math. Comput. 72 (2003), 1577-1595. Zbl 1031.65133, MR 1986794, 10.1090/S0025-5718-02-01502-8 |
| Reference: | [7] Bacuta, C., Bramble, J. H., Xu, J.: Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains.J. Numer. Math. 11 (2003), 75-94. Zbl 1050.65108, MR 1987589, 10.1515/156939503766614117 |
| Reference: | [8] Bramble, J. H., Scott, R.: Simultaneous approximation in scales of Banach spaces.Math. Comput. 32 (1978), 947-954. Zbl 0404.41005, MR 0501990, 10.1090/S0025-5718-1978-0501990-5 |
| Reference: | [9] Costabel, M., Dauge, M., Nicaise, S.: Mellin analysis of weighted Sobolev spaces with nonhomogeneous norms on cones.Around the Research of Vladimir Maz'ya. I. Function Spaces International Mathematical Series (New York) 11. Springer, New York (2010), 105-136. Zbl 1196.46024, MR 2723815, 10.1007/978-1-4419-1341-8_4 |
| Reference: | [10] Costabel, M., Dauge, M., Nicaise, S.: Analytic regularity for linear elliptic systems in polygons and polyhedra.Math. Models Methods Appl. Sci. 22 (2012), Article ID 1250015, 63 pages. Zbl 1257.35056, MR 2928103, 10.1142/S0218202512500157 |
| Reference: | [11] Costabel, M., Stephan, E.: Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation.Mathematical Models and Methods in Mechanics Banach Center Publications 15. PWN, Warsaw (1985), 175-251. Zbl 0655.65129, MR 0874845, 10.4064/-15-1-175-251 |
| Reference: | [12] Costabel, M., Stephan, E., Wendland, W. L.: On boundary integral equations of the first kind for the bi-Laplacian in a polygonal domain.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 10 (1983), 197-241. Zbl 0563.45007, MR 0728434 |
| Reference: | [13] Dahlke, S.: Besov regularity for elliptic boundary value problems in polygonal domains.Appl. Math. Lett. 12 (1999), 31-36. Zbl 0940.35064, MR 1751404, 10.1016/S0893-9659(99)00075-0 |
| Reference: | [14] Dahlke, S., DeVore, R. A.: Besov regularity for elliptic boundary value problems.Commun. Partial Differ. Equations 22 (1997), 1-16. Zbl 0883.35018, MR 1434135, 10.1080/03605309708821252 |
| Reference: | [15] Dauge, M.: Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions.Lecture Notes in Mathematics 1341. Springer, Berlin (1988). Zbl 0668.35001, MR 0961439, 10.1007/BFb0086682 |
| Reference: | [16] DeVore, R. A., Lorentz, G. G.: Constructive Approximation.Grundlehren der Mathematischen Wissenschaften 303. Springer, New York (1993). Zbl 0797.41016, MR 1261635, 10.1007/978-3-662-02888-9 |
| Reference: | [17] Ebmeyer, C.: Mixed boundary value problems for nonlinear elliptic systems with $p$-structure in polyhedral domains.Math. Nachr. 236 (2002), 91-108. Zbl 1147.35309, MR 1888558, 10.1002/1522-2616(200203)236:1<91::AID-MANA91>3.0.CO;2-1 |
| Reference: | [18] Ebmeyer, C., Frehse, J.: Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains.Math. Nachr. 203 (1999), 47-74. Zbl 0934.35048, MR 1698636, 10.1002/mana.1999.3212030104 |
| Reference: | [19] Evans, L. C.: Partial Differential Equations.Graduate Studies in Mathematics 19. AMS, Providence (2010). Zbl 1194.35001, MR 1625845, 10.1090/gsm/019 |
| Reference: | [20] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order.Grundlehren der Mathematischen Wissenschaften 224. Springer, Berlin (1983). Zbl 0562.35001, MR 0737190, 10.1007/978-3-642-61798-0 |
| Reference: | [21] Grisvard, P.: Singularities in Boundary Value Problems.Recherches en Mathématiques Appliquées 22. Springer, Berlin (1992). Zbl 0766.35001, MR 1173209 |
| Reference: | [22] Grisvard, P.: Elliptic Problems in Nonsmooth Domains.Classics in Applied Mathematics 69. SIAM, Philadelphia (2011). Zbl 1231.35002, MR 3396210, 10.1137/1.9781611972030 |
| Reference: | [23] Jerison, D. S., Kenig, C. E.: The Neumann problem in Lipschitz domains.Bull. Am. Math. Soc., New Ser. 4 (1981), 203-207. Zbl 0471.35026, MR 0598688, 10.1090/S0273-0979-1981-14884-9 |
| Reference: | [24] Jerison, D. S., Kenig, C. E.: Boundary value problems on Lipschitz domains.Studies in Partial Differential Equations MAA Studies in Mathematics 23. Mathematical Association of America, Washington (1982), 1-68. Zbl 0529.31007, MR 0716504 |
| Reference: | [25] Jerison, D., Kenig, C. E.: The inhomogeneous Dirichlet problem in Lipschitz domains.J. Funct. Anal. 130 (1995), 161-219. Zbl 0832.35034, MR 1331981, 10.1006/jfan.1995.1067 |
| Reference: | [26] Kondrat'ev, V. A.: Boundary value problems for elliptic equations in domains with conical or angular points.Trudy Moskov. Mat. Obšč. 16 (1967), 209-292 Russian. Zbl 0162.16301, MR 0226187 |
| Reference: | [27] Kozlov, V. A., Maz'ya, V. G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities.Mathematical Surveys and Monographs 52. AMS, Providence (1997). Zbl 0947.35004, MR 1469972, 10.1090/surv/052 |
| Reference: | [28] Maz'ya, V. G., Plamenevskij, B. A.: The coefficients in the asymptotics of solutions of the elliptic boundary value problem in domains with conical points.Math. Nachr. 76 (1977), 29-60 Russian. Zbl 0359.35024, MR 601608, 10.1002/mana.19770760103 |
| Reference: | [29] Maz'ya, V. G., Plamenevskij, B. A.: Estimates in $L_p$ and in Hölder classes and the Miranda-Agmon maximum principle for the solutions of elliptic boundary value problems in domains with singular points on the boundary.Transl., Ser. 2, Am. Math. Soc. 123 (1984), 1-56 translation from Math. Nachr. 81 1978 25-82. Zbl 0554.35035, MR 492821, 10.1090/trans2/123 |
| Reference: | [30] Maz'ya, V. G., Rossmann, J.: Elliptic Equations in Polyhedral Domains.Mathematical Surveys and Monographs 162. AMS, Providence (2010). Zbl 1196.35005, MR 2641539, 10.1090/surv/162 |
| Reference: | [31] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations.Cambridge University Press, Cambridge (2000). Zbl 0948.35001, MR 1742312 |
| Reference: | [32] Melenk, J. M.: On Generalized Finite-Element Methods: Ph. D. Thesis.University of Maryland, College Park (1995). MR 2692949 |
| Reference: | [33] Nazarov, S. A., Plamenevsky, B. A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries.De Gruyter Expositions in Mathematics 13. Walter de Gruyter, Berlin (1994). Zbl 0806.35001, MR 1283387, 10.1515/9783110848915 |
| Reference: | [34] Nicaise, S.: Polygonal Interface Problems.Methoden und Verfahren der Mathematischen Physik 39. Peter Lang, Frankfurt am Main (1993). Zbl 0794.35040, MR 1236228 |
| Reference: | [35] Rojik, C.: $p$-Version Projection-Based Interpolation: Ph. D. Thesis.Technische Universität Wien, Wien (2019). 10.34726/hss.2019.65840 |
| Reference: | [36] Savaré, G.: Regularity results for elliptic equations in Lipschitz domains.J. Funct. Anal. 152 (1998), 176-201. Zbl 0889.35018, MR 1600081, 10.1006/jfan.1997.3158 |
| Reference: | [37] Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces.Lecture Notes of the Unione Matematica Italiana 3. Springer, Berlin (2007). Zbl 1126.46001, MR 2328004, 10.1007/978-3-540-71483-5 |
| Reference: | [38] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators.Johann Ambrosius Barth, Heidelberg (1995). Zbl 0830.46028, MR 1328645 |
| Reference: | [39] Triebel, H.: Function spaces in Lipschitz domains and on Lipschitz manifolds: Characteristic functions as pointwise multipliers.Rev. Mat. Complut. 15 (2002), 475-524. Zbl 1034.46033, MR 1951822, 10.5209/rev_REMA.2002.v15.n2.16910 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
