This paper examines when it is possible to find a smooth potential on a C1 domain D with prescribed normal derivatives at the boundary. It is shown that this is always possible when D is a Liapunov-Dini domain, and this restriction on D is essential. An application concerning C1 superharmonic extension is given.

Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces. As an application of the boundedness of the maximal operator, we establish Sobolev's inequality for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. We are also concerned with Trudinger's inequality and the continuity for Riesz potentials.

Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{ℍ,\nu }f$ and Riesz potentials $I_{ℍ,\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $ℍ$. We also obtain Sobolev type inequalities for a $C^1$ function on $ℍ$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t)=t^{p\left(x\right)}+{\left(b\left(x\right)t\right)}^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions, $p\left(x\right)<q\left(x\right)$ for $x\in ℍ$, and $b(·)$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r\le d$) where $(r,\theta ,z)$ denotes the cylindrical co-ordinates in ${ℝ}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, 1) $\mathrm{dist}(0z,\partial A)=O\left(k^{1/2}\right)$; $\mathrm{diam}A=O(exp(-c_0{k}^{3/2}))$; 2) ${\left(k^{1/2}\Psi \right)}_{k\in \mathbb{N}}$ converges to a vortex cylinder $U_m$ (see...