We investigate the transport equation ${\partial }_{t}u(t,x)+b(t,x)·D_{x}u(t,x)=0$. Our result improves the classical criteria of uniqueness of weak solutions in the case of irregular coefficients: b ∈ BV, $di{v}_{x}b\in BMO$. To obtain our result we use a procedure similar to DiPerna and Lions’s one developed for Sobolev vector fields. We apply renormalization theory for BV vector fields and logarithmic type inequalities to obtain energy estimates.

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation ${\partial }_{t}u+\frac{.}{\dot{}}\left(bu\right)=0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $divA(x,u,Du)+B(x,u,DU)=0$, under natural polynomial growth of the coefficient functions $A$ and $B$. We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $-div\left(a\right(x,u,\nabla u\left)\right)+g(x,u)=f-divF$ in Ω, where Ω is a bounded open domain of ${ℝ}^N$, N ≥ 2 and $a:\Omega \times ℝ\times {ℝ}^N\to {ℝ}^N$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to ${\prod }_{i=1}^N{L}^{p^{\text{'}}\left(·\right)}\left(\Omega \right)$.

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div(x,\nabla u)+{a\left(x\right)\left|u\right|}^{p-2}u={g\left(x\right)\left|u\right|}^{p-2}u+h\left(x\right){\left|u\right|}^{s-1}u$ in ${ℝ}^N$ ⎨ ⎩ u > 0, $li{m}_{\left|x\right|\to \infty }u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider...

The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain $\Omega$ whose boundary $\partial \Omega$ is formed by two circles ${\Gamma }_1$, ${\Gamma }_2$ with the same center $S_0$ and radii $R_1$, $R_2=R_1+\varrho$, where $\varrho \ll R_1$. On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for $u=0$ are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

We consider functionals of a potential energy ${\mathbb{P}}_{\psi }\left(u\right)$ corresponding to $\mathrm{𝑎𝑛}\mathrm{𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}-\mathrm{𝑣𝑎𝑙𝑢𝑒}\mathrm{𝑝𝑟𝑜𝑏𝑙𝑒𝑚}$. We are dealing with $𝑎\mathrm{𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛}\mathrm{𝑜𝑓}𝑎\mathrm{𝑡ℎ𝑖𝑛}\mathrm{𝑎𝑛𝑛𝑢𝑙𝑎𝑟}\mathrm{𝑝𝑙𝑎𝑡𝑒}$ with $\mathrm{𝑁𝑒𝑢𝑚𝑎𝑛𝑛}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}\mathrm{𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠}$. Various types of the subsoil of the plate are described by various types of the $\mathrm{𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒}$ nonlinear term $\psi \left(u\right)$. The aim of the paper is to find a suitable computational algorithm.

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-{div\left(\right|x|}^{-ap}{\left|\nabla u\right|}^{p-2}{\nabla u)+h\left(x\right)|u|}^{p-2}u=g\left(x\right){\left|u\right|}^{r-2}u$, x ∈ Ω, ⎨ ⎩ ${\left|x\right|}^{-ap}{\left|\nabla u\right|}^{p-2}\partial u/\partial \nu =\lambda f\left(x\right){\left|u\right|}^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in ${ℝ}^N$, that is, $\Omega ={ℝ}^N\setminus D$, where D is a bounded domain in ${ℝ}^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as $$-\mathrm{div}a(x,\nabla u)=b(x,\nabla u)+\mu$$ where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$, $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^1$ function, we prove a uniqueness result under more restrictive hypotheses on the operator.

We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1,2}\left({ℝ}^n\right)$. This criterion consists of comparing the value of a functional $\int f\left(u\right)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$, and we prove that the rate of convergence to zero is connected to regularity: $u\in W^{1,2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to...

In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

We study nonlinear diffusion problems of the form $u_t=u_{xx}+f\left(u\right)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f\left(u\right)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f\left(u\right)$ which is $C^1$ and satisfies $f\left(0\right)=0$, we show that the omega limit set $\omega \left(u\right)$ of every bounded positive solution is determined by a stationary...