Abstract
In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type
Initially, we prove a partial regularity result with the method of A-polycaloric approximation, which is a parabolic analogue of the harmonic approximation lemma of De Giorgi. Moreover, we prove better estimates for the maximal parabolic Hausdorff-dimension of the singular set of weak solutions, using fractional parabolic Sobolev spaces. Thereby, we also consider different situations, which yield a better dimension reduction result, including the low dimensional case and coefficients A(z, D m u), independent of the lower order derivatives of u.
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Acerbi E. and Mingione G. (2001). Regularity results for a class of functionals with non-standard growth. Arch. Rat. Mech. Anal. 156: 121–140
Acerbi E. and Mingione G. (2007). Gradient estimates for a class of parabolic systems. Duke Math. J. 136: 285–320
Acerbi E., Mingione G. and Seregin G.A. (2004). Regularity results for parabolic systems related to a class of non-newtonian fluids. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21(1): 25–60
Adams R.A. (1978). Sobolev Spaces. Academic Press, New York
Bögelein, V.: Regularity results for weak and very weak solutions of higher order parabolic systems. Ph.D.Thesis (2007)
Bojarski B. and Iwaniec T. (1983).Analytical foundations of the theory of quasiconformal mappings in \({\mathbb R^n}\). Ann. Acad. Sci. Fenn. Ser. A I 8: 257–324
Campanato S. (1966). Equazioni paraboliche del secondo ordine e spazi \({\fancyscript {L}^{2,\theta} (\Omega,\delta)}\). Ann. Mat. Pura Appl. IV Ser. 73: 55–102
Campanato S. (1967). Maggiorazioni interpolatorie negli spazi \({H_\lambda^{m,p}(\Omega)}\). Ann. Mat. Pura Appl. 75: 261–276
Campanato S. (1982). Differentiability of the solutions of nonlinear elliptic systems with natural growth. Ann. Mat. Pura Appl. 131(4): 75–106
Campanato S. (1984). On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. Ann. Mat. Pura Appl. 137(4): 83–122
Campanato S. and Cannarsa P. (1980). Differentiability and partial Hölder continuity of the solutions of non-linear elliptic systems of order 2m with quadratic growth. Ann. Scuola Norm. Sup. Pisa 8: 285–309
Da Prato G. (1965). Spazi \({\fancyscript {L}^{(p,\vartheta)}(\Omega,\delta)}\) e loro proprieta. Ann. Mat. Pura Appl. IV. Ser. 69: 383–392
De Giorgi, E.: Frontiere orientate di misura minima. Sem. Scuola Normale Superiore Pisa (1960–1961)
Domokos A. (2004). Differentiability of solutions for the non-degenerate p-laplacian in the heisenberg group. J. Differ. Equ. 204: 439–470
Duzaar F., Gastel A. and Grotowski J.F. (2001). Optimal partial regularity for nonlinear elliptic systems of higher order. J. Math. Sci., Tokyo 8(3): 463–499
Duzaar F. and Grotowski J.F. (2000). Optimal interior partial regularity for nonlinear elliptic systems: the method of a-harmonic approximation. Manuscr. Math. 103: 267–298
Duzaar F. and Mingione G. (2004). The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Part. Differ. Equ. 20: 235–256
Duzaar F. and Mingione G. (2004). Regularity for degenerate elliptic problems via p-harmonic approximation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21: 735–766
Duzaar F. and Mingione G. (2005). Second order parabolic systems, optimal regularity and singular sets of solutions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22: 705–751
Duzaar, F., Mingione, G., Steffen, K.: Second order parabolic systems with p-growth and regularity (to appear, 2008)
Fefferman C. and Stein E.M. (1972). Hp spaces of several variables. Acta Math. 129: 137–193
Föglein, A.: Regularität von Lösungen gewisser Systeme elliptischer partieller Differentialgleichungen in der Heisenbarg-Gruppe. Diplomarbeit (2005)
Giaquinta M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton
Giaquinta, M.: Introduction to regularity theory for nonlinear elliptic systems (1993)
Giaquinta M. and Struwe M. (1982). On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179: 437–451
Giusti E. (2003). Direct Methods in the Calculus f Variations. World Scientific Publishing Company, Tuck Link, Singapore
John J. and Nirenberg L. (1961). On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14: 415–426
Kinnunen J. and Lewis J.L. (2000). Higher integrability for parabolic systems of p-laplacian type. Duke Math. J. 102: 253–271
Ladyženskaja O.A., Solonnikov V.A. and Ural’ceva N.N. (1968). Linear and Quasilinear Equations of Parabolic Type, vol. 23. American Mathematical Society, Providence
Mingione G. (2003). Bounds for the singular set of solutions to non linear elliptic systems. Calc. Var. Partial Differ. Equ. 18(4): 373–400
Mingione G. (2003). The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166: 287–301
Nirenberg L. (1960). On elliptic partial differential equations. Ann. Sc. Norm. Super., Pisa III. Ser. 123: 115–162
Simon J. (1987). Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. IV. Ser. 146: 65–96
Simon, L.: Lectures on Geometric Measure Theory. Proc. Centre Math. Anal., Austr. Nat. Univ., Canberra (1983)
Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Lectures in Math., ETH Zrich, Birkhuser, Basel (1996)
Stredulinsky E.W. (1980). Higher integrability from reverse Hölder inequalities. Indiana Univ. Math. J. 29: 407–413
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Bögelein, V. Partial regularity and singular sets of solutions of higher order parabolic systems. Annali di Matematica 188, 61–122 (2009). https://doi.org/10.1007/s10231-008-0067-4
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DOI: https://doi.org/10.1007/s10231-008-0067-4