Papers

We consider on Riemannian manifolds the Trudinger equation \begin{equation*}\partial _{t}u=\Delta _{p}u^{\frac{1}{p-1}},\end{equation*} where $p>1$. We prove that a relative $p$–Faber–Krahn inequality is equivalent to the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions of Trudinger's equation. We also derive an improved long-time upper estimate under a uniform $p$–Faber–Krahn inequality.

We study the long-time behavior of nonnegative solutions to a nonlinear diffusion equation with gradient absorption on infinite weighted graphs. The equation combines a discrete p-Laplacian diffusion term with a nonlinear absorption term of Hamilton-Jacobi type, modeling processes where both diffusion and nonlinear damping occur on discrete structures. Under the assumptions that the graph satisfies a polynomial volume growth condition and that the parameters satisfy certain ordering relations, we establish sharp decay estimates for the total mass of the solution. We prove that the total mass decays to zero as time tends to infinity whenever the absorption exponent lies below a critical threshold. This critical exponent depends explicitly on the volume growth rate of the graph and the diffusion exponent, and coincides with the Fujita-type threshold known for analogous equations in the continuous Euclidean setting. Our proof combines Hardy and Hölder inequalities with a carefully chosen time-dependent scaling parameter adapted to the graph geometry. The results are new even in the linear diffusion case and extend naturally to a broad class of graphs, including integer lattices and Cayley graphs of finitely generated nilpotent groups.

Ischemic heart diseases represent the leading cause of mortality worldwide. Revascularization, the process to restore blood flow in blockages, shows promise. To this end, mathematical models for angiogenesis, the process by which new blood vessels form from existing ones, have been extremely well investigated. In the current work, we consider a classical two species model for angiogenesis, consisting of cell and VEGF populations. However, we assume the cells move according to p-Laplacian diffusion, which could be both “fast" ($1<p<2$) and “slow" ($p>2$), in addition to normal diffusion ($p=2$). We first show that the system is well posed in a weak sense when $p>\frac{3}{2}$, for sufficiently small initial data. Next, we show that the p-Laplacian can lead to several novel dynamics not reported earlier; these include increased cellular proliferation via bi-modal and multi spike solutions, gain of regularity, prevention of finite time blow-up, cell depletion via finite time extinction, and Turing patterns. We discuss applications of these results for cardiac health via a digital twins framework.

We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schrödinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schrödinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.

We investigate the analysis and phase ordering dynamics of the active Cahn–Hilliard equation, providing novel results beyond the current state of the art concerning the well-posedness and the characterization of its coarsening dynamics. We consider both regular polynomial and singular logarithmic potentials. In particular, we exploit a new method based on heteroclinic trajectories in the phase plane to characterize static kink profiles and spherical droplet states, recovering the exact values of key quantities related to static phase-separated configurations; moreover, we develop a theory accounting for surface tension modifications driven by activity and local interface curvature, which explains the power-law shift $L(t)\sim t^{\frac{1}{z}}$ from $z=3$ to $z=4$ induced by activity for the characteristic domain length conjectured in the literature. This shows that there is a transitory effect before the attainment of a finite saturation length. We also design an efficient numerical scheme, based on finite elements, to approximate the model, proving its well-posedness and stability both for regular and singular potentials. In dimensions $d=2,3$ with singular potential, the convergence analysis of the finite element approximation proves the local-in-time existence and uniqueness of weak solutions satisfying the physical constraint $\phi \in (-1, 1)$. In dimension $d=1$ with singular potential, we establish global-in-time well-posedness and regularity of weak solutions under a smallness condition on the activity parameter. Finally, we show numerical simulations for different test cases which prove that our numerical algorithm correctly reproduces the expected phase separation dynamics. Moreover, we show the results for coarsening dynamics at late times which present a power law shift from $z=3$ to $z=4$ prior to reaching late-time length saturation, which confirms our theoretical findings.

It is well known that the computational realization of Gibbs probability measures, $e^{-\mathbb{U}(\mathbf{x})}/Z$, plays a central role in sampling and optimization. In this paper, we introduce two types of dynamics that exhibit rapid convergence towards these Gibbs measures. The mechanism driving this rapid convergence is the enhanced dissipation associated with these transport-diffusion dynamics. Motivated by these enhanced dynamics, we design numerical algorithms for sampling from the target Gibbs measure. Finally, we provide the corresponding particle systems that may yield other effective numerical samplers.

In this article, we establish Schauder-type estimates for the logarithmic Laplacian. We show that for a $\kappa$-Hölder inhomogeneous term, the solution is also $\kappa$-Hölder in the interior. In fact, the interior regularity slightly exceeds $\kappa$-Hölder smoothness up to a logarithmic correction. Additionally, we prove a Harnack inequality for non-negative solutions.

Let $G$ be any given continuous positive function on $\mathbb{R}_+$. Let $V$ be radial with $|V(x)|\leq G(|x|)$. We prove a Landis-type theorem for any real-valued solution of $\Delta u=Vu$ on $\mathbb{R}^n$. We construct a decay threshold $e^{-g(r)}$, where $g$ is a strictly increasing function which can be computed explicitly in terms of $G$. Under suitable assumptions the exponent in the decay threshold is proportional to the Agmon distance associated with $G$.

For integers \(n,\ell\geq1\), set \(Q_\ell=2(n+\ell)+2\) and \(q_\ell=2Q_\ell/(Q_\ell-2)\). On the Siegel domain \(\Ucal=\Heis^n\times(0,\infty)\), we study \[ -\Delta_{\Heis^n}v -4\rho\bigl(v_{\rho\rho}+T^2v\bigr)-4\ell v_\rho =|v|^{q_\ell-2}v. \] For its Dirichlet form \(E_\ell\), we determine the sharp Sobolev constant and all extremals, prove \[ S_\ell\|v\|_{q_\ell}^2\leq E_\ell(v),\qquad E_\ell(v)-S_\ell\|v\|_{q_\ell}^2 \geq\kappa_{n,\ell} \operatorname{dist}_{\dot S^1_\ell} \bigl(v,\mathfrak M_\ell^{\R}\bigr)^2, \] where \(\mathfrak M_\ell^{\R}\) is the extremal cone. We classify nonnegative finite-energy solutions and establish the linearized kernel, profile decomposition, and attainment of the optimal stability quotient. Cayley conjugation yields ground-state symmetry and nondegeneracy, global Palais–Smale compactness, and perturbative existence for critical equations on \(\CHyp^{n+1}\). The mechanism is the radial lift \(v^\uparrow(z,w,t)=v(z,t,|w|^2)\), which reverses the interior-to-boundary construction by realizing the Siegel domain as a symmetry-reduced slice of a larger Heisenberg group. The completed-space reduction resolves the degenerate axis, hidden auxiliary concentration, and the mismatch of extremal cones. Together with the Cayley transform, this supplies the missing nonlinear layer—classification, stability, bubbling, and variational compactness—on complex hyperbolic space and provides a blueprint for other rank-one symmetric spaces.

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

In this manuscript, we investigate an inverse boundary value problem for a reaction-diffusion-convection system in a bounded domain of $\mathbb{R}^{1+n}$, $n\geq 2$. We aim to obtain a stability estimate for determining the time-dependent convection coefficient and matrix-valued potential from boundary measurements represented by the Dirichlet-to-Neumann map. We consider a partial data setting in which the measurements are available only on a subset of the lateral boundary that slightly exceeds one-half of the boundary. We first establish the well-posedness of the associated initial-boundary value problem. Subsequently, by combining Carleman estimates with suitable geometric optics solutions, we derive stability estimates for the unknown coefficients. More precisely, we prove a double logarithmic ($\log$-$\log$) stability estimate for the time-dependent convection coefficient from the knowledge of the partial Dirichlet-to-Neumann map. This stability result is then employed to recover the matrix-valued potential, yielding a triple logarithmic ($\log$-$\log$-$\log$) stability estimate for the zeroth-order coefficient.

In this paper, we consider identification of space dependent coefficients in a time dependent PDE, which is non-autonomous, due to a bilinear control term. We prove linearized uniqueness from time trace observations, in particular also considering non-local observations in the form of weighted integrals of the state over he spatial domain. The result is obtained in an abstract setting and applied, first of all to the identification of a potential in a diffusion equation for illustration purposes, second to the reconstruction of equilibrium magnetization, relaxation rates and field inhomogeneity as space dependent quantities in model based qunatitative magnetic resonance imaging with the Bloch-Torrey equation, which constitutes the real world application motivating this study.

This paper is concerned with the large-time behavior of quasilinear Fokker-Planck equations with confinement on the whole space $\mathbb{R}^d$. It aims at characterizing all relative entropy functionals such that the entropy method à la Bakry-Émery yields exponential convergence of all solutions towards the unique steady state (with the same mass as the initial condition). We call such entropies admissible. The convergence rate is determined by the uniform convexity parameter of the confinement potential. As such, this program extends the analogous study of linear Fokker-Planck equations [Bakry-Émery, Arnold-Markowich-Toscani-Unterreiter] to the nonlinear case, and it derives additional functionals for the nonlinear case — beyond the Ralston-Newman entropies used in [Jüngel-Carrillo-Markowich-Toscani-Unterreiter]. Two key results are the characterization of those nonlinear Fokker-Planck equations which admit all entropy functionals that are admissible for the corresponding linear Fokker-Planck equation, and vice versa, the characterization of all admissible entropies for a given nonlinearity. The latter quest for power-law nonlinearities yields a large family of entropies for the porous-medium equations, but only the Ralston-Newman entropy for the fast-diffusion equations. Additional results include the derivation of new generalized Csiszár-Kullback and generalized Log-Sobolev inequalities for our entropy functionals as well as moment-weighted $L^1$–convergence estimates for the Fokker-Planck solutions.

Motivated by classical constitutive inequalities in isotropic nonlinear elasticity theory, we investigate the monotonicity of isotropic tensor functions of the form \[ \Sigma_f\colon\mathrm{Sym}(n)\to\mathrm{Sym}(n)\,,\quad \Sigma_f(Q^T\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)\, Q) = Q^T\mathrm{diag}(f(\lambda_1,\dotsc,\lambda_n))\, Q \quad\forall\;Q\in\mathrm{O}(n) \] with a vector function $f=(f_1,\dotsc,f_n)\colon\mathbb{R}^n\to\mathbb{R}^n$ which is symmetric, i.e.\ satisfies \[ f_i(\lambda_{\pi(1)},\dotsc,\lambda_{\pi(n)}) = f_{\pi(i)}(\lambda_1,\dotsc,\lambda_n) \] for any permutation $\pi\colon\{1,\dotsc,n\}\to\{1,\dotsc,n\}$, where $\mathrm{Sym}(n)$ denotes the space of symmetric $n\times n$ matrices, $\mathrm{O}n$ is the orthogonal group and $\mathrm{diag}(\lambda_1,\dotsc,\lambda_n)$ is the diagonal matrix with diagonal entries $\lambda_1,\dotsc,\lambda_n\in\mathbb{R}$. We prove that vector-monotonicity of $f$ on $\mathbb{R}^n$ is equivalent to matrix-monotonicity of the induced isotropic tensor function $\Sigma_f$ on $\mathrm{Sym}(n)$. Our results generalize the Chandler Davis theorem for convex scalar isotropic functions and are obtained independently of Hill's original proof of this equivalence. We also discuss simple invertibility conditions for isotropic matrix functions. We conclude by showing that injectivity of the Cauchy stress $V\mapsto\sigma(V)$, continuous differentiability, and positive definiteness of $\mathrm{sym}\,\mathrm D\sigma(1\!\!\!\:1)$ in the natural state imply the strong Baker-Ericksen inequalities.

We consider weak solutions of the second-order elliptic equation $\operatorname{div} (\mathbf{M}\nabla u) = \operatorname{div}\mathbf{F}$ in $\Omega\subset {\mathbb R}^d$ with Dirichlet boundary conditions, where $\mathbf{M}$ is a uniformly elliptic real-valued symmetric matrix $\mathbf{M}:\Omega \rightarrow \mathbb{R}^{d\times d}$ such that $\frac{1}{K} |\xi|^2 \leq\langle \mathbf{M}\xi,\xi \rangle \leq K |\xi|^2, \forall \xi \in \mathbb{R}^d, K > 1$. We prove that $\|\nabla u\|_p \leq C\|\mathbf{F}\|_p$ for any $p\in \left(\frac{26K-16}{13K-3}, \frac{26K-16}{13K-13}\right)$.

The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

In this work, we conduct a comprehensive study of problem \begin{equation*} \begin{cases} -\Delta_1 u + g(u)|Du| = h(u)f & \text{in }\Omega, u=0 & \text{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $f\in L^1(\Omega)$ is a nonnegative datum, and $g,h$ are nonnegative continuous functions on $(0,\infty)$ that may be singular at the origin. Under the minimal assumptions that $g$ is integrable near zero and $h$ is bounded at infinity, we explore the existence of a global $BV(\Omega)$ solution. Furthermore, a comparison principle is proved under suitable monotonicity assumptions on $h$. This framework avoids any growth restrictions on $h$ near the origin, thus allowing for highly singular terms. To handle these nonlinearities, we introduce a novel approach that takes advantage of the rigid structure of the 1-Laplacian operator.

We study the spectral stability of periodic shear flows for the two-dimensional Navier–Stokes equations on the $\beta$-plane in the long-wave regime. While it is known that non-rotating periodic shear flows are generically unstable to sufficiently long-wave perturbations, our results show that planetary rotation suppresses this instability mechanism. Using a perturbative analysis based on Kato's reduction, we derive an asymptotic expansion for the principal eigenvalue of the linearized operator that is uniform in the Rossby parameter $\beta \geq 0$, and establish an explicit stability criterion in terms of the shear profile, the viscosity, and the Rossby parameter. Under the critical scaling where the Rossby parameter $\beta$ and the rescaled longitudinal wavenumber $\varepsilon$ are of comparable size, this criterion extends Yudovich's classical long-wave instability threshold to rotating flows and reveals a sharp transition between stability and instability governed by the range of the ratio $\beta/\varepsilon$. These results provide a rigorous analysis of how viscosity, shear, and planetary rotation interact to determine long-wave stability on the $\beta$-plane.

We consider the defocusing nonlinear Schrödinger equation for step-like data connecting two, in general different, genus-one periodic states. Starting from the Jost solutions associated with the left and right backgrounds, we determine the corresponding scattering data and reorganize them through a scalar conjugation and a suitable symmetrization. The resulting formulation identifies the inverse problem with a full dark-soliton gas Riemann–Hilbert problem supplemented by a radiative jump on the real axis. In this way, the effects of the two periodic backgrounds are separated, and a precise link is established between step-like periodic scattering and the full-gas construction. Assuming the absence of discrete eigenvalues and appropriate regularity of the scattering data, we establish the existence and uniqueness of the associated Riemann–Hilbert problem.

In this paper we apply the method of Kosygina, Rezakhanlou and Varadhan (CPAM 2006) to establish stochastic homogenization of Hamilton-Jacobi-Bellman (HJB) equation with a vanishing non-local integro-differential operator and a convex super-linearly growing Hamiltonian in stationary ergodic random medium. Their method, first designed for homogenization of HJB equations with vanishing Laplacian operator, relies on stochastic optimal control representation of the solution, a key construction of approximate super-correctors and the technique of linking diffusion process to abstract diffusion in random media. We show how the procedures can be carried out for HJB equations with jump-diffusion. In particular, for the construction of approximate super-correctors, we represent the non-local integro-differential operator as the divergence of a regular integral operator acting on the gradient.

We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo–Nirenberg inequalities, including the BMO endpoint case. Furthermore, we establish endpoint asymptotic results as $s\to0^+$ and $s\to1^-$, proving that the asymptotic factors appearing in these inequalities have optimal order. Under the additional assumption that the underlying function space is rearrangement invariant, we show that the optimal Gagliardo–Nirenberg target spaces are precisely those given by the Calderón–Lozanovskiĭ space. This completely characterizes the rearrangement invariant target spaces for which the corresponding Gagliardo–Nirenberg inequalities hold, thereby answering an open question posed by K. Leśnik, T. Roskovec, and F. Soudský. These results can be applied to various function spaces; in particular, they are completely new in the off-diagonal and BMO cases.

By working in a setting of azimuthal and $z$-directional invariance, we reduce the existence of electrically neutral, magnetically driven, equilibria of the two-species Vlasov-Maxwell (VM) equation to a second-order ODEs in the radial variable. In contrast to previous works, we consider specific situations that correspond physically to $z$-pinch, $\theta$-pinch, and screw-pinch equilibria in plasma physics. Semi-explicit solutions in these situations are provided.

In this paper, we study the axisymmetric Hall magnetohydrodynamics where $u = u^re_r+u^ze_z$ and $B = B^\theta e_\theta$. We construct small, smooth, compactly supported, and axisymmetric initial data $(u_{0},B_{0})$ and prove that there is a unique smooth axisymmetric solution with norm inflation in $H^{\sigma-1}(\mathbb R^3) \times H^{\sigma}(\mathbb R^3)$ for $1<\sigma<7/2$. Moreover, we study the case with fractional dissipation $(-\Delta)^\beta u$ and $(-\Delta)^\alpha B$, modifying the initial data slightly and proving that the solution inflates in $H^{\sigma-1}(\mathbb R^3) \times H^{\sigma}(\mathbb R^3)$ for $1<\sigma<7/2-2\max\{\alpha,\beta\}$ and $ 0 \le \alpha, \beta < 5/4$.

We study the Fujita phenomenon for semilinear heat inequalities generated by variable-speed Laplacians on infinite weighted graphs. Assuming that the graph carries a proper adapted path metric, we establish an integral volume-growth criterion forcing every nonnegative global classical supersolution on the open cylinder $(0,\infty)\times V$ to vanish, without assuming an initial value or trace. We prove that a nontrivial supersolution of this kind exists if and only if the equation has a positive global Cauchy solution for some nonzero point-source datum. A complementary heat-kernel construction gives global Cauchy solutions for all sufficiently small point-source data when the same volume integral converges and a matching anchored heat-kernel upper bound is available. This proves sharpness on integer lattices and on a family of logarithmically perturbed weighted half-lines; in the latter examples, even the exponent of an iterated logarithm can determine the existence–nonexistence alternative. A finer nonexistence criterion couples intrinsic volume growth with the capacity of intrinsic annuli. Its proof combines parabolic testing, a Laplace–resolvent reduction, and a pathwise decomposition of resolvent currents. The nonexistence results require no volume-doubling property, Poincaré inequality, heat-kernel bound, or stochastic completeness.

We study the finite time blow up phenomenon for the quintic, mass critical gKdV equation. We prove the existence of a class of solutions $U$ with infinite point, finite time blow up behavior, at the particular blow up rate $$\|\partial_x U(t)\|_{L^2} \sim (T-t)^{-\nu}\quad \text{as} \quad t \uparrow T,$$ where $\nu = \frac 12$, $T$ is the blow up time and where the travel speed of the blow up bubble is logarithmic. Therefore, we call this behaviour slowly travelling infinite point blow up. The special blow up rate $\nu =\frac 12$ is a threshold which separates finite and infinite point bubbling. In a previous work arXiv:2511.13538, the author constructed other infinite point blow up solutions for the continuum of rates $\nu\in(\frac 12,1)$, using polynomial tails in the space variable and extending the results in arXiv:1209.2510, restricted to $\nu> \frac{11}{13}$. However, that work suggested a change of the tail for the threshold case. In the present paper, we consider an exponentially decaying tail on the right in space. As in arXiv:2511.13538, the initial data can be taken arbitrarily close to the ground state in $H^1$. From a technical perspective, in addition to the change of tail, we have to adapt the energy-virial functional to the presence of the exponential tail, by modifying a scaling term used to control the right-hand side of the blow up solution.

Two nonradial scattering criteria are established for the three-dimensional focusing energy-critical generalized Hartree equation below the ground-state threshold, under a bounded-scale concentration–compactness reduction. The first criterion is formulated in terms of fixed-center occupation windows and quantitative bounds on the motion of the concentration center. The second is based on uniform low-frequency \(L^2\)-decay, which yields finite mass, precompactness in the inhomogeneous energy space, a zero-momentum normalization, and sublinear center drift. The proofs combine a localized Hartree virial identity with a critical Hardy–Littlewood–Sobolev estimate controlling the nonlocal tail.

We study the initial-boundary value problem for the Cahn-Hilliard equation with degenerate mobility and singular diffusion at pure phases. This model describes the dynamics of phase separation in polymer blends with associated Flory-Huggins-de Gennes free energy. We prove the existence of global weak solutions in three-dimensional bounded and smooth domains, assuming that the initial datum has finite energy. We show the classical regularity in $L^2(0,T;H^2(\Omega))$ for both the solution $u$ and the function $\phi(u)=\arcsin(u)$ through entropy estimates. In addition, when $\Omega$ is convex, a new elliptic-type argument yields the refined regularity $u, \phi(u) \in L^4(0,T; H^2(\Omega))$.

We prove that the planar Dacorogna–Marcellini energy $f_\gamma(A)=|A|^4-2\gamma|A|^2\det A$ is quasiconvex exactly when it is rank-one convex, i.e. if and only if $|\gamma|\leq\frac{2}{\sqrt3}$. The proof uses a monotonicity property of the energy functional along the componentwise heat flow. As a corollary of our method, we show that for homogeneous quartic polynomials on $2 \times 2$ matrices invariant by left and right rotation, quasiconvexity is equivalent to rank one convexity.

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.

This paper reproduces the front matter — preface, overview and table of contents — of a monograph by the author, submitted for publication under the title Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory. The book isolates a class of non-linear operators, which we call Kantorovich operators, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf–Lax–Oleinik semi-group. Having no adjoint, the duality between such an operator and measures is carried instead via a convex functional on pairs of probability distributions — a source and a target — which we call a skew-linear entropy, and which is a general form of optimal mass transport. The extensive overview reproduced here describes the resulting ergodic theory, in which minimal measures, a Mather constant, weak KAM solutions and an Aubry set are attached to an arbitrary Kantorovich operator, extending Aubry–Mather theory well beyond its origins in Hamiltonian dynamics.