Optimal \(L^{p}\)- \(L^{q}\)-estimates for parabolic boundary value problems with inhomogeneous data. (English) Zbl 1210.35066

Denk, Robert; Hieber, Matthias; Prüss, Jan

Math. Z. 257, No. 1, 193-224 (2007).

Summary: We investigate vector-valued parabolic initial boundary value problems \({(\mathcal A(t,x,D)}\), \({\mathcal B_j(t,x,D))}\) subject to general boundary conditions in domains \(G\) in \({\mathbb R^n}\) with compact \(C^{2 m}\)-boundary. The top-order coefficients of \({\mathcal A}\) are assumed to be continuous. We characterize optimal \(L^{p}\)- \(L^{q}\)-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on \({\mathcal A}\) and the Lopatinskii-Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) are necessary for these \(L^{p}\)- \(L^{q}\)-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.

MSC:

35J40	Boundary value problems for higher-order elliptic equations
42B15	Multipliers for harmonic analysis in several variables

Keywords:

parabolic boundary value problems with general boundary conditions; vector-valued Sobolev spaces

References:

[1]	Agmon S. (1962). On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math. 15: 119–147 · Zbl 0109.32701
[2]	Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959) II. Comm. Pure Appl. Math. 17, 35–92 (1964) · Zbl 0093.10401
[3]	Agranovich M., Vishik M.I. (1964). Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surv. 19: 53–157 · Zbl 0137.29602
[4]	Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function spaces, Differential Operators and Nonlinear Analysis, pp. 9–126, Teubner-Texte Math 133, Teubner, Stuttgart (1993) · Zbl 0810.35037
[5]	Amann H. (1995). Linear and Quasilinear Parabolic Problems. Birkhäuser, Boston · Zbl 0819.35001
[6]	Amann H. (2001). Elliptic operators with infinite-dimensional state spaces. J. Evol. Equ. 1: 143–188 · Zbl 1018.35023
[7]	Amann H., Hieber H., Simonett G. (1994). Bounded H calculus for elliptic operators. Diff. Integral Equ. 7: 613–653 · Zbl 0799.35060
[8]	Auscher P., Hofmann S., Lewis J., Tchamitichian P. (2001). Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators. Acta Math. 187(2): 161–190 · Zbl 1163.35346
[9]	Burkholder D.L. (1986). Martingales and Fourier analysis in Banach spaces. In: Letta, G., Pratelli, M. (eds) Probability and Analysis, Lecture Notes in Mathematics, vol. 1206, pp 61–108. Springer, Berlin · Zbl 0605.60049
[10]	Clément Ph., Prüss J. (2001). An operator-valued transference principle and maximal regularity on vector-valued L p -spaces. In: Lumer, G., Weis, L. (eds) Evolution Equation and Applied Physical Life Sciences, Lecture Notes in Pure Appllied Mathematics, vol. 215, pp 67–87. Marcel Dekker, New York · Zbl 0988.35100
[11]	Cowling M., Doust I, . McIntosh A., Yagi A. (1996). Banach space operators with a bounded H functional calculus. J. Aust. Math. Soc. Ser. A 60: 51–89 · Zbl 0853.47010
[12]	Denk, R., Hieber, M., Prüss, J.: \({\mathcal R}\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 788 (2003) · Zbl 1274.35002
[13]	Denk R., Dore G., Hieber M., Prüss J., Venni V. (2004). New thoughts on old results of R. T. Seeley. Math. Ann. 328: 545–583 · Zbl 1113.35057
[14]	Escher J., Prüss J., Simonett G. (2003). Analytic solutions for the Stefan problem with Gibbs-Thomson correction. Reine Angew. Math. 563: 1–52 · Zbl 1242.35220
[15]	Duong, X.T.: H functional calculus of second order elliptic partial differential operators on L p spaces. In: Doust, I., Jefferies, B., Li, C., McIntosh, A. (eds.) Miniconference on Operators in Analysis. Proc. Centre Math. Anal. A.N.U. vol. 24, pp. 91–102 (1989)
[16]	Duong X.T., McIntosh A. (1996). Functional calculi for second order elliptic partial differential operators with bounded measurable coefficients. J. Geom. Anal. 6: 181–205 · Zbl 0897.47041
[17]	Duong X.T., Robinson D. (1996). Semigroup kernels, Poisson bounds and holomorphic functional calculus. J. Funct. Anal. 142(1): 89–128 · Zbl 0932.47013
[18]	Duong X.T., Simonett G. (1997). H calculus for elliptic operators with nonsmooth coefficients. Diff. Integral Equ. 10: 201–217 · Zbl 0892.47017
[19]	Duong X.T., Yan L. (2002). Bounded holomorphic functional calculus for non-divergence form operators. Diff. Integral Equ. 15: 709–730 · Zbl 1020.47033
[20]	Grisvard P. (1972). Spaci di trace e applicazioni. Rend. Math. 5: 657–729 · Zbl 0272.46025
[21]	Haller, R., Heck, Hieber, M.: L p -L q -estimates for parabolic systems in non-divergence form with VM0-coefficients. J. London Math. Soc. (to appear) · Zbl 1169.35339
[22]	Haller R., Heck H., Noll A. (2002). Mikhlin’s theorem for operator-valued Fourier multipliers in n variables. Math. Nachr. 244: 110–130 · Zbl 1054.47013
[23]	Heck H., Hieber M. (2003). Maximal L p -regularity for elliptic operators with VMO-coefficients. J. Evol. Equ. 3: 332–359 · Zbl 1225.35080
[24]	Hieber M., Prüss J. (1997). Heat-kernels and maximal L p -L q -estimates for parabolic evolution equations. Comm. Partial Diff. Equ. 22: 1647–1669 · Zbl 0886.35030
[25]	Hieber M., Prüss J. (1998). Functional calculi for linear operators in vector-valued L p -spaces via the transference principle. Adv. Diff. Equ. 3: 847–872 · Zbl 0956.47008
[26]	Kalton W., Lancien G. (2000). A solution to the problem of L p -maximal regularity. Math. Z. 235: 559–568 · Zbl 1010.47024
[27]	Kalton W., Weis L. (2001). The H calculus and sums of closed operators. Math. Ann. 321: 319–345 · Zbl 0992.47005
[28]	Kunstmann, P.: Maximal L p -regularity for second order elliptic operators with uniformly continuous coefficients on domains. In: Progress in Nonlinear Differential Equations and Their Applications, vol. 55. pp. 293–305 Birkäuser (2003) · Zbl 1230.35033
[29]	Kunstmann P., Weis L. (2004). Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H functional calculus. Lect. Notes Math. 1855: 65–311 · Zbl 1097.47041
[30]	Prüss J. (2002). Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in L p -spaces. Math. Bohem. 127: 311–327 · Zbl 1010.35064
[31]	Prüss J., Sohr H. (1993). Imaginary powers of elliptic second order differential operators in L p spaces. Hiroshima Math. J. 23: 161–192 · Zbl 0790.35023
[32]	Sobolevskii P.E. (1975). Fractional powers of coercively positive sums of operators. Soviet Math. Dokl. 16: 1638–1641 · Zbl 0333.47010
[33]	Solonnikov, V.A.: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Fust. Steklov 83, 3–163 (1965) (Russian). Engl. Transl.: Proc. Steklov Inst. Math. 83, 1–184 (1965) · Zbl 0164.12502
[34]	Strkalj, Z., Weis, L.: On operator-valued Fourier multiplier theorems. Trans. Am. Math. Soc. (to appear) · Zbl 1209.42005
[35]	Triebel, H.: Interpolation Theory, Function spaces, Differential Operators. North-Holland (1978) · Zbl 0387.46032
[36]	Triebel H. (1992). Theory of Function Spaces II. Birkhäuser Verlag, Basel · Zbl 0763.46025
[37]	Weidemaier P. (2002). Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed L p .norm. Electr Res. Announc. Am. Math. Soc. 8: 47–51 · Zbl 1015.35036
[38]	Weis L. (2001). Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann. 319(4): 735–758 · Zbl 0989.47025