# Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the following: In the book "Linear and Quasilinear Equations of Parabolic Type" by Ladyzenskaja and others the following initial value problem is studied: Let $\Omega \subset \mathbb{R}^n$ be a domain and $S$ be its boundary. Furthermore we denote $\Omega_T:= \Omega \times [0, T)$ and $S_T:= S \times [0, T)$. We then study the initial value problem $$Lu=f, u\mid_{S_T}=0, u \mid_{t=0}= \psi_0(x) \, \, (*)$$ where $L$ is an operator of the form $Lu=\frac{\partial u}{\partial t}-\frac{\partial }{\partial x_i}(a_{ij}u_{x_i}+a_iu)-b_iu_{x_i}-au$, with $a_{ij}(x, t)$ satisfying the inequality $v \sum_{i=1}^n \zeta_i^2 \leq a_{ij} \zeta_i \zeta_j \leq \mu \sum_{i=1}^n \zeta_i^2$. One now has to introduce a concept of weak solutions for this equation: We start with some definitions. We denote by $W^{1, 0}_2(Q_T)$ the Hilbert space with scalar product $$(u, v)_{W^{1, 0}(Q_T)}= \int_{Q_T} (uv+ u_{x_k}v_{x_k}) dx dt$$ and by $V_2(Q_T)$ the Banach space consisting of all elements of $W^{1, 0}_2(Q_T)$, which have a finite norm $$\vert u \vert_{Q_T}= \mbox{ess sup}_{0 \leq t \leq T} \left( \int_{\Omega} \vert u \vert^q dx \right)^{\frac{1}{q}}+ \left( \int_{Q_T} u_x^2 \right)^{\frac{1}{2}}$$ Moreover we let $\mathring{V_2}(Q_T)$ be the closure of the smooth functions with compact support in $V_2(Q_T)$. \ Now we retun to the problem (): We define a weak solution of () to be a function $u$ from $\mathring{V}_2(Q_T)$ which fulfills for almost all $t_1$ the identity $$I(t_1; u, \eta)= \int_{\Omega} \psi_0(x) \eta(x, 0) ds$$ where we let $$I(t_1; u, \eta)= \int_{\Omega} u(x, t_1) \eta(x, t_1) dx- \int_0^{t_1} \int_{\Omega} u \eta_t dx dt+ \int_0^{t_1} (L_1(u, \eta)+ L_2(, \eta)) dt=0$$ with the abbreviations $$L_1(u, \eta)= \int_M ((a_{ij}u_{x_j}+ a_iu)\eta_{x_i}+ (b_iu_{x_i}+au)\eta)dx$$ and $$L_2(f, \eta)=\int_M (f_i\eta_{x_i}+ f \eta)dx$$ Then it is shown in the book "Linear and Quasilinear Equations of Parabolic Type" by Ladyzenskaja and others that the problem (*) always has a weak solution which furthermore is contained in $C^{2+\alpha, 1+\frac{\alpha}{2}}(Q_T)$ provided that the coefficients $a_{ij}$, $b_i$, $c$ as well as the free terms $f$ and $\psi_0$ are elements of $C^{\alpha, \frac{\alpha}{2}}(Q_T)$ (this is precisely Theorem 12.1 in Chapter iii). But what I can not find in the book is a statement about the regularity at the boundary, i.e. whether $u$ has to be continuous at the boundary or whatever. \ I thus would like to know: Can somebody say anything about the regularity at the boundary? Is it continuous at least?

• Anybody. who knows an answer – user86084 Feb 3 '16 at 6:13
• You could look at Lieberman's book. – username Mar 13 '18 at 16:42
• – username Mar 13 '18 at 16:49