We prove an existence theorem of Cauchy-Kovalevskaya type for the equation $D_{t}u(t,z)=f(t,z,u\left({\alpha }^{\left(0\right)}(t,z)\right),D_{z}u\left({\alpha }^{\left(1\right)}(t,z)\right),...,D_z^{k}u\left({\alpha }^{\left(k\right)}(t,z)\right))$ where f is a polynomial with respect to the last k variables.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified...

We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.

We consider the local projection finite element method for the discretization of a scalar convection-diffusion equation with a divergence-free convection field. We introduce a new fluctuation operator which is defined using an orthogonal $L^2$ projection with respect to a weighted $L^2$ inner product. We prove that the bilinear form corresponding to the discrete problem satisfies an inf-sup condition with respect to the SUPG norm and derive an error estimate for the discrete solution.

We prove some multiplicity results concerning quasilinear elliptic equations with natural growth conditions. Techniques of nonsmooth critical point theory are employed.

We consider the problem of minimizing the energy$$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$among all functions $u\in SB{V}^2\left(\Omega \right)$ for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for $E(·)$ is proved and the asymptotic behaviour of the solutions is investigated.

We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.