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On the continuity of solutions to degenerate elliptic equations. (English) Zbl 1213.35231

The very interesting paper under review deals with the local behaviour of weak solutions to the divergence form equation \[ \text{div\,} A(x)\nabla u=0\qquad \text{in}\;\Omega\subset\mathbb R^n \] with a real and symmetric matrix \(A(x)\) of measurable entries satisfying \[ w(x)|\xi|^2\leq \big\langle A(x)\xi,\xi\big\rangle\leq v(x)|\xi|^2 \] with suitable weights \(w(x)\geq0\) and \(v(x)\).
In the special case of \(A_2\)-Muckenhoupt weights satisfying \(v(x)=\Lambda w(x)\) a.e. in \(\Omega\) with \(\Lambda\geq1,\) S. Chanillo and R. L. Wheeden [Commun. Partial Differ. Equations 11, 1111–1134 (1986; Zbl 0634.35035)] proved continuity of the weak solution under additional, very strong assumptions on the oscillations \(w(x)\) and \(v(x)\).
In the paper under review, the authors consider the case \(w(x) = 1/K(x)\) and \(v(x) = K(x)\in A_2.\) Assuming \(K\) to satisfy the Gehring \(G_n\)-condition \[ G_n(K)=\sup_B { {\left({1\over{|B|}}\int_B K(x)^ndx\right)^{1/n}} \over{{1\over{|B|}}\int_B K(x)dx} }<\infty \] with the supremum taken over all balls \(B\subset \mathbb R^n,\) and setting \(S_v\) for the domain of the maximal function of \(v\), \(S_v =\{x\in\Omega:\;Mv(x)<\infty\}\), it is proved that the restriction to \(S_v\) of the precise representative \(\tilde u\) of any non-negative solution \(u\) is continuous.

MSC:

35J70 Degenerate elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
35D30 Weak solutions to PDEs

Citations:

Zbl 0634.35035
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References:

[1] Alvino, A.; Trombetti, G., Sulle migliori costanti di maggiorazione per una classe di equazioni ellittiche degeneri, Ric. Mat., 27, 413-428 (1978) · Zbl 0403.35027
[2] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), Oxford University Press · Zbl 0957.49001
[3] Astala, K.; Gill, J.; Rohde, S.; Saksman, E., Optimal regularity for planar mappings of finite distortion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 1-19 (2010) · Zbl 1191.30007
[4] Astala, K.; Iwaniec, T.; Martin, G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser. (2009), University Press: University Press Princeton · Zbl 1182.30001
[5] Buckley, S. M., Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc., 340, 253-272 (1993) · Zbl 0795.42011
[6] Chanillo, S.; Wheeden, R. L., Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math., 107, 5, 1191-1226 (1985) · Zbl 0575.42026
[7] Chanillo, S.; Wheeden, R. L., Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 11, 10, 1111-1134 (1986) · Zbl 0634.35035
[8] Fabes, E. B.; Kenig, C. E.; Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7, 77-116 (1982) · Zbl 0498.35042
[9] Fiorenza, A.; Krbec, M., On the domain and range of the maximal operator, Nagoya Math. J., 158, 43-61 (2000) · Zbl 1039.42015
[10] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics Math. (2001), Springer-Verlag: Springer-Verlag Berlin, reprint of the 1998 edition · Zbl 0691.35001
[11] Giusti, E., Metodi diretti nel calcolo delle variazioni (1994), Unione Matematica Italiana: Unione Matematica Italiana Bologna · Zbl 0942.49002
[12] Iwaniec, T.; Migliaccio, L.; Moscariello, G.; Passarelli di Napoli, A., A priori estimates for nonlinear elliptic complexes, Adv. Differential Equations, 8, 5, 513-546 (2003) · Zbl 1290.35074
[13] Iwaniec, T.; Sbordone, C., Quasiharmonic fields, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, 5, 519-572 (2001) · Zbl 1068.30011
[14] Johnson, R.; Neugebauer, C. J., Homeomorphisms preserving \(A_p\), Rev. Mat. Iberoam., 3, 2, 249-273 (1987) · Zbl 0677.42019
[15] Johnson, R.; Neugebauer, C. J., Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwend., 12, 1, 3-11 (1993) · Zbl 0777.42004
[16] Korey, M. B., Ideal weights: Asymptotically optimal version of doubling, absolute continuity, and bounded mean oscillation, J. Fourier Anal. Appl., 4, 4-5, 491-519 (1998) · Zbl 0943.42008
[17] Kruzkov, S. N., Certain properties of solutions to elliptic equations, Soviet Math., 4, 686-695 (1963)
[18] Migliaccio, L., Some characterizations of Gehring \(G_p\)-class, Houston J. Math., 19, 1, 89-95 (1993) · Zbl 0778.42013
[19] Murthy, M. K.V.; Stampacchia, G., Boundary value problems for degenerate elliptic operators, Ann. Mat. Pura Appl., 80, 1, 1-122 (1968) · Zbl 0185.19201
[20] Onninen, J.; Zhong, X., Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6, 1, 103-116 (2007) · Zbl 1150.35055
[21] Trudinger, N. S., On the regularity of generalized solutions of linear non-uniformly elliptic equations, Arch. Ration. Mech. Anal., 42, 51-62 (1971) · Zbl 0218.35035
[22] Wik, I., On Muckenhoupt’s classes of weight functions, Studia Math., 94, 245-255 (1989) · Zbl 0686.42012
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