We show the $L^1$ contraction and comparison principle for weak (and, more generally, renormalized) solutions of the elliptic–parabolic problem $j(v)_t-\text{div}(\nabla w+F(w))=f(t,x)$, $w=\varphi(v)$ in $(0,T)\times\varOmega\subset \mathbb{R}^+\times\mathbb{R}^N$ with inhomogeneous Dirichlet boundary datum $g\in L^2(0,T;W^{1,2}(\varOmega))$ for $w$ (the boundary datum is taken in the sense $w-g\in L^2(0,T;H^{1}_0(\varOmega))$) and initial datum $j_o\in L^1(\varOmega)$ for $j(v)$. Here $\varphi$ and $j$ are non-decreasing, and we assume that $F$ is just continuous.

Our proof consists in doubling of variables in the interior of $\varOmega$ as introduced by Carrillo in 1999, and in a careful treatment of the flux term near the boundary of $\varOmega$. For the latter argument, the result is restricted to the linear dependence on $\nabla w$ of the diffusion term. The proof allows for a wide class of domains $\varOmega$, including, for example, weakly Lipschitz domains with Lipschitz cracks.

We obtain the corresponding results for the associated stationary problem and discuss on generalization of our technique to the case of nonlinear diffusion operators.

We establish that, for $n\geq3$, the elliptic equation

$$ -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u $$

on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all $\lambda>0$ provided that

$$ \mu,\nu>-2,\quad\max\bigg\{2,\frac{n+2\mu+2}{n-2}\bigg\}<q<\frac{2(n+\mu)}{n-2}\quad\text{and}\quad p=\frac{2(n+\nu)}{n-2}. $$

When $q=2$ and $n>2\mu+6$, the same result holds for $\lambda>0$ small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation

$$ -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, $$

on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all $\tilde\lambda>0$ provided that $a,\nu>-2$ and

$$ \max\bigg\{2,\frac{n+a-\sqrt{\bar{\chi}-\chi}}{\sqrt{\bar{\chi}}}\bigg\} <q<\frac{n+a}{\sqrt{\bar{\chi}}}\quad\text{with~}\bar{\chi}=\bigg(\frac{n-2}{2}\bigg)^2. $$

When $q=2$, $n>2a+6$ and $0<\chi<\bar\chi-(a+2)^2$, the same result holds for $\tilde\lambda>0$ small.

We find positive rapidly decaying solutions for the equation

$$ -\text{div}(K(x)\nabla u)=K(x)u^{2^*-1}+\lambda K(x)|x|^{\alpha-2}u $$

in $\mathbb{R}^N$, where $N\geq3$, the nonlinearity is given by the critical Sobolev exponent $2^*=2N/(N-2)$, the weight is $K(x)=\exp(\tfrac14|x|^\alpha)$, $\alpha\geq2$ and $\lambda$ is a parameter.

We study the asymptotic behaviour of two nonlinear eigenvalue problems which involve $p$-Laplacian-type operators. In the first problem we consider the limit as $p\to\infty$ of the sequences of the $k$th eigenvalues of the $p$-Laplacian operators. The second problem we study is the homogenization of nonlinear eigenvalue problems for some $p$-Laplacian-type operators with $p$ fixed. Our asymptotic analysis relies on a convergence result for particular critical values of a class of Rayleigh quotients, stated in a unified framework, and on the notion of $\varGamma$-convergence.

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. We solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular, we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore, we construct examples in which the genericity assumption does not hold and indicate that the situation is different for these.

The action of the symmetric group $S_4$ on the tetrahedron algebra, introduced by Hartwig and Terwilliger, is studied. This action gives a grading of the algebra which is related to its decomposition into a direct sum of three subalgebras isomorphic to the Onsager algebra. The ideals of both the tetrahedron algebra and the Onsager algebra are determined.

In a real Hilbert space $H$, we study the bifurcation points of equations of the form $F(\lambda,u)=0$, where $F:\mathbb{R}\times H\rightarrow H$ is a function with $F(\lambda,0)=0$ that is Hadamard differentiable, but not necessarily Fréchet differentiable, with respect to $u$ at $u=0$. In this context, there may be bifurcation at points $\lambda$ where $D_{u} F(\lambda,0):H\rightarrow H$ is an isomorphism. We formulate some additional conditions on $F$ that ensure that bifurcation does not occur at a point where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. Then, in the case where $F(\lambda,\cdot)$ is a gradient, we give conditions that imply that bifurcation occurs at a point $\lambda$. These conditions may be satisfied at points where $D_{u}F(\lambda,0):H\rightarrow H$ is an isomorphism. We demonstrate the use of these abstract results in the context of nonlinear elliptic equations of the form

$$ -\Delta u(x)+q(x)u(x)=\lambda\eta(x)^{-1}f(\eta(x)u(x)) $$

and

$$ -\Delta u(x)+q(x)u(x)-\eta(x)^{-1}f(\eta(x)u(x))=\lambda u(x), $$

where $u\in H^{2}(\mathbb{R}^{N})$.

Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

We derive geometrical information on smooth surfaces in $\mathbb{R}^4$ from the geometry of their images under linear projections to 3-spaces.

We characterize a topological convex space $C$ in terms of the family $\mathcal{A}(C)$ of real continuous affine functions on $C$. Our main result states that two topological convex spaces $C_1$ and $C_2$ are affine-homeomorphic if and only if $\mathcal{A}(C_1)$ and $\mathcal{A}(C_2)$ are isomorphic as ordered unital vector spaces.