an:01192428
Zbl 0907.35026
Caffarelli, Luis A.; Kenig, Carlos E.
Gradient estimates for variable coefficient parabolic equations and singular perturbation problems
EN
Am. J. Math. 120, No. 2, 391-439 (1998).
00049245
1998
j
35B45 35K20 35B25 35B50 35B65
Dini continuous coefficients; monotonicity formula approach; smooth dirichlet data; Hopf maximum principle
It is studied the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form, with bounded lower order terms and Dini continuous coefficients of the type
\[
\text{div } A(x,t)\nabla u-\partial_tu+b_i(x,t){\partial u\over\partial x_i}(x,t)+c(x,t)u(x)=f(x,t)+ \text{div }\vec g(x,t).
\]
Uniform spatial Lipschitz estimates are established for some singular perturbation problems. It is used a monotonicity formula approach to be shown that Dini continuity of the coefficients at a given point yields boundedness of the gradient at the same point. It is obtained a modulus of continuity estimate up to the boundary for gradients of solutions with smooth Dirichlet data. A compactness argument and a judicious application of Harnack's principle allow to obtain a Hopf maximum principle for the considered class of equations.
Lubomira Softova (Sofia)