Geodesic
A note on Campanato's $L^p$-regularity with continuous coefficients
Eurasian mathematical journal, Tome 13 (2022) no. 4, pp. 44-53.

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In this note we consider local weak solutions of elliptic equations in variational form with data in $L^p$. We refine the classical approach due to Campanato and Stampacchia and we prove the $L^p$-regularity for the solutions assuming the coefficients merely continuous. This result shows that it is possible to prove the same sharp $L^p$-regularity results that can be proved using classical singular kernel approach also with the variational regularity approach introduced by De Giorgi. This method works for general operators: parabolic, in nonvariational form, of order $2m$. 
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C. Bernardini; V. Vespri; M. Zaccaron. A note on Campanato's $L^p$-regularity with continuous coefficients. Eurasian mathematical journal, Tome 13 (2022) no. 4, pp. 44-53. http://geodesic.mathdoc.fr/item/EMJ_2022_13_4_a2/

[1] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011 | MR | Zbl

[2] A. P. Calderón, A. Zygmund, “On the existence of certain singular integrals”, Acta Math., 88 (1952), 85–139 | DOI | MR | Zbl

[3] A. P. Calderón, A. Zygmund, “On singular integrals”, Amer. J. Math., 78 (1956), 289–309 | DOI | MR | Zbl

[4] S. Campanato, “Equazioni ellittiche del secondo ordine e spazi $\mathcal{L}^{2,\lambda}$”, Ann. Mat. Pura Appl., 69 (1965), 321–382 | DOI | MR

[5] S. Campanato, “Su un teorema di interpolazione di G. Stampacchia”, Ann. Scuola Norm. Sup. Pisa, 20 (1965), 649–652 | MR

[6] S. Campanato, Sistemi ellittici in forma divergenza. Regolarità all-interno, Quaderni Scuola Norm. Sup. Pisa, Pisa, 1980, 187 pp. | MR | Zbl

[7] S. Campanato, G. Stampacchia, “Sulle maggiorazioni in $L^p$ nella teoria delle equazioni ellittiche”, Boll. Un. Mat. Ital., 20 (1965), 393–399 | MR | Zbl

[8] P. Cannarsa, B. Terreni, V. Vespri, “Analytic semigroups generated by non-variational elliptic systems of second order under Dirichlet boundary conditions”, J. Math. Anal. Appl., 112 (1985), 56–103 | DOI | MR | Zbl

[9] E. De Giorgi, “Sulla differenziabilità e l-analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3:3 (1957), 25–43 (Italian) | MR | Zbl

[10] E. De Giorgi, “Un esempio di estremali discontinue per un problema variazionale di tipo ellittico”, Boll. UMI, 4 (1968), 135–137 | MR

[11] E. Di Benedetto, Degenerate parabolic equations, Universitext book series, Springer Verlag, Berlin-New York, 1993 | DOI | MR

[12] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998, xviii+662 pp. | MR | Zbl

[13] D. Hilbert, “Mathematische Probleme”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1900 (1900), 253–297

[14] T. Iwaniec, “Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators”, Studia Math., 75 (1983), 293–312 | DOI | MR | Zbl

[15] N. V. Krylov, M. V. Safonov, “A property of the solutions of parabolic equations with measurable coefficients”, Izv. Akad. Nauk SSSR Ser. Mat., 44:1 (1980), 161–175 (in Russian) | MR | Zbl

[16] O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear parabolic equations, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1968 | DOI | MR

[17] G. Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, 181, Second edition, American Mathematical Society, Providence, RI, 2017 | DOI | MR | Zbl

[18] G. Mingione, “Nonlinear aspects of Calderón-Zygmund theory”, Jahresber. Dtsch. Math. Ver., 112:3 (2010), 159–191 | DOI | MR | Zbl

[19] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–954 | DOI | MR | Zbl

[20] J. Schauder, “Uber lineare elliptische Differentialgleichungen zweiter Ordnung”, Mathematische Zeitschrift, 38 (1934), 257–282 | DOI | MR | Zbl

[21] J. Schauder, “Numerische Abschätzungen in elliptischen linearen Differentialgleichungen”, Studia Mathematica, 5 (1937), 34–42 | DOI

[22] G. Stampacchia, “The spaces $\mathcal{L}^{p,\lambda}$, $N^{p,\lambda}$ and interpolation”, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 443–462 | MR | Zbl

[23] G. Stampacchia, “$\mathcal{L}^{p,\lambda}$-spaces and interpolation”, Comm. Pure Appl. Math., 17 (1964), 293–306 | DOI | MR | Zbl

[24] V. Vespri, “Analytic semigroups generated by ultraweak operators”, Proc. Royal Soc. Edinburgh Sect. A, 119 (1991), 87–105 | DOI | MR | Zbl