Geodesic
Time dependent boundary norms for kernels and regularizing properties of the double layer heat potential
Eurasian mathematical journal, Tome 8 (2017) no. 1, pp. 76-118.

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We introduce a class of norms for time dependent kernels on the boundary of Lipschitz parabolic cylinders and we prove theorems of joint continuity of integral operators upon variation of both the kernel and the density function. As an application, we prove that the integral operator associated to the double layer heat potential has a regularizing property on the boundary. 
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M. Lanza de Cristoforis; P. Luzzini. Time dependent boundary norms for kernels and regularizing properties of the double layer heat potential. Eurasian mathematical journal, Tome 8 (2017) no. 1, pp. 76-118. http://geodesic.mathdoc.fr/item/EMJ_2017_8_1_a7/

[1] K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, etc., 1985 | MR | Zbl

[2] F. Dondi, M. Lanza de Cristoforis, “Tangential derivatives of the double layer potential of second order elliptic differential operators with constant coefficients”, Memoirs on Differential Equations and Mathematical Physics, 71 (2017) (to appear)

[3] M. Costabel, “Boundary integral operators for the heat equation”, Integral Equations Operator Theory, 13:4 (1990), 498–552 | DOI | MR | Zbl

[4] G. B. Folland, Real analysis. Modern techniques and their applications, Second edition, John Wiley Sons, Inc., New York, 1999 | MR

[5] M. Gevrey, “Sur les équations aux dérivées partielle du type parabolique”, Journal de mathématiques pures et appliquées, 9 (1913), 305–471

[6] M. Gevrey, “Sur les équations aux dérivées partielle du type parabolique (suite)”, Journal de mathématiques pures et appliquées, 10 (1914), 105–148 | Zbl

[7] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Second edition, Springer-Verlag, Berlin, 1983 | MR | Zbl

[8] S. Hofmann, J. L. Lewis, “$L^2$ solvability and representation by caloric layer potentials in time-varying domains”, Ann. of Math., 144:2 (1996), 349–420 | DOI | MR | Zbl

[9] L. I. Kamynin, “On the smoothness of thermal potentials”, Differencial'nye Uravnenija, 1 (1965), 799–839 (Russian) | MR | Zbl

[10] L. I. Kamynin, “On the smoothness of thermal potentials. II. Thermal potentials on the surface of type $L_{1,1,(1+\alpha)/2}^{1,\alpha,\alpha/2}$”, Differencial'nye Uravnenija, 2 (1966), 647–687 (in Russian) | MR | Zbl

[11] L. I. Kamynin, “On the smoothness of thermal potentials. V. Thermal potentials $U$, $V$ and $W$ on surfaces of type $\Pi^{m+1,\alpha,\alpha/2}_{2m+1,1,(1+\alpha)/2}$ and $\Pi^{m+1,1,(1+\alpha)/2}_{2m+3,1,\alpha,\alpha/2}$”, Differensial'nye Uravnenija, 4 (1968), 347–365 (in Russian) | MR | Zbl

[12] L. I. Kamynin, “On the smoothness of thermal potentials. V. Thermal potentials $U$, $V$ and $W$ on surfaces of type $L^{m+1,\alpha,\alpha/2}_{2m+1,1,(1+\alpha)/2}$ and $L^{m+1,1,(1+\alpha)/2}_{2m+3,\alpha,\alpha/2}$. II”, Differencial'nye Uravnenija, 4 (1968), 881–895 (in Russian) | MR

[13] R. Kress, Linear integral equations, Applied Mathematical Sciences, 82, Springer-Verlag, Berlin, 1989 | DOI | MR | Zbl

[14] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, 12, American Mathematical Society, Providence, RI, 1996 | DOI | MR | Zbl

[15] O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1968 (in Russian) | MR

[16] J. L. Lewis, M. A. M. Murray, The method of layer potentials for the heat equation in time-varying domains, Mem. Amer. Math. Soc., 114, no. 545, 1995 | MR

[17] P. Luzzini, Derivate tangenziali del potenziale di doppio strato calorico, Tesi di laurea magistrale, relatore M. Lanza de Cristoforis, Università degli studi di Padova, 2015, 86 pp.

[18] J. Schauder, “Potentialtheoretische Untersuchungen”, Math. Z., 33:1 (1931), 602–640 | DOI | MR

[19] J. Schauder, “Bemerkung zu meiner Arbeit “Potentialtheoretische Untersuchungen I (Anhang)””, Math. Z., 35:1 (1933), 536–538 | DOI | MR

[20] Van Tun, “Theory of the heat potential. I. Level curves of heat potentials and the inverse problem in the theory of the heat potential”, Ž. Vyčisl. Mat. i Mat. Fiz., 4 (1964), 660–670 (in Russian) | MR

[21] Van Tun, “Theory of the heat potential. II. Smoothness of contour heat potentials”, Ž. Vyčisl. Mat. i Mat. Fiz., 5 (1965), 474–487 (in Russian) | MR

[22] Van Tun, “A theory of the thermal potential. III. Smoothness of a plane thermal potential”, Ž. Vyčisl. Mat. i Mat. Fiz., 5 (1965), 658–666 (in Russian) | MR