A linear parabolic equation with special singularities is studied. A priori boundary estimates are established for the maximum of the modulus of the gradient of a solution and for the Hölder constants as well. These estimates depend linearly on the functions appearing on the right-hand side of the equation. Bibliography: 7 titles.
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D. E. Apushinskaya and A. I. Nazarov,Estimates on the Boundary of a Domain for the Hölder Norms of the Derivatives of Solutions to General Quasilinear Elliptic and Parabolic Equations, Prepr. LOMI, P-I-85, Leningrad (1985).
O. A. Ladyzhenskaya and N. N. Ural'tseva, “Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality,”Proc. Steklov Inst. Math., No. 2, 109–135 (1989).
G. M. Lieberman, “The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data,”Commun. Partial Differ. Equations,11, No. 2, 167–229 (1986).
G. M. Lieberman, “The first initial-boundary value problem for quasilinear second order parabolic equations,”Ann. Sci. Norm. Super. Pisa, Cl. Sci., IV,13, No. 3, 347–387 (1986).
A. I. Nazarov and N. N. Ural'tseva, “Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation,”Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova,147, 95–109 (1985).
N. B. Krylov, “Estimates of the maximum of the solution of a parabolic equation and estimates of the distribution of a semimartingale,”Mat. Sb.,130(172), No. 2(6), 207–221 (1986).
O. A. Ladyzhenskaya and N. N. Ural'tseva, “Estimates of the Hölder constant for functions satisfying a uniformly elliptic or uniformly parabolic quasilinear inequality with unbounded coefficients,”Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova,147, 72–94 (1985).
Translated fromProblemy Matematicheskogo Analiza, No. 14, 1995, pp. 3–27.
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Apushkinskaya, D.E., Nazarov, A.I. Boundary estimates for the first-order derivatives of a solution to a nondivergent parabolic equation with composite right-hand side and coefficients of lower-order derivatives. J Math Sci 77, 3257–3276 (1995). https://doi.org/10.1007/BF02364860
- Parabolic Equation
- Boundary Estimate
- Special Singularity
- Linear Parabolic Equation
- Nondivergent Parabolic Equation