We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega$. We prove the global $C^{1,\alpha }$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1,\alpha }$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

Let $Q_T$ be a cylinder in ${\mathbb{R}}^{n+1}$ and $x=\left(x^{\prime },t\right)\in {\mathbb{R}}^n\times \mathbb{R}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$\left\{\begin{array}{cc}{u}_t-\stackrel{n}{\sum _{i,j=1}}{a}^{ij}\left(x\right)D_{ij}u=f\left(x\right)\hfill & \text{q.o. in }{Q}_T,\hfill \\ u\left(x\right)=0\hfill & \text{su }\partial Q_T,\hfill \end{array}\right.$$ in the Morrey spaces $W_{p,\lambda }^{2,1}\left(Q_T\right)$, $p\in \left(1,\mathrm{\infty }\right)$, $\lambda \in \left(0,n+2\right)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

Let $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ be a linear partial differential operator with holomorphic coefficients, where$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\text{'}}}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m\>k$$and$$z=(z_0,z^{\text{'}})\in C^{n+1}.$$We consider Cauchy problem with holomorphic data$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\text{'}})={\widehat{u}}_i(z^{\text{'}}\left)\right(0\le i\le k-1).$$We can easily get a formal solution $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\text{'}}\left)\right(z_0{)}^n/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial }_z)$, there is a function $u_S\left(z\right)$ holomorphic except on $\{z_0=0\}$ such that $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ and $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ as $z_0\to 0$ in $S$.

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

As a model for elliptic boundary value problems, we consider the Dirichlet problem for an elliptic operator. Solutions have singular expansions near the conical points of the domain. We give formulas for the coefficients in these expansions.

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foiaş, Guillopé and Temam.