Papers

We investigate the sharp regularity up to the boundary for viscosity solutions of fully nonlinear parabolic equations with oblique derivative boundary conditions given by \begin{equation*} \left\{ \begin{array}{rclcl} F(D^2u,x,t) - u_{t} &=& f(x,t) & \text{in} & Q_{1}^{+}, \\ \beta(x,t) \cdot Du &=& g(x,t) & \text{on} & Q_{1}^{*}. \end{array} \right. \end{equation*} The regularity theory is developed according to the integrability of the source term, the smoothness of the boundary data, and the oscillation of the coefficients of the operator \(F\), using a compactness method combined with polynomial approximation. In the borderline case \(f\in L^{n+2}\), we obtain Log-Lipschitz continuity of solutions up to the boundary. Under stronger integrability, specifically when \(f\in L^{p}\) for some \(p>n+2\), we establish optimal \(C^{1+\alpha',\,\frac{1+\alpha'}{2}}\) boundary estimates. At a higher regularity level, we prove Schauder-type estimates under appropriate assumptions on the operator \(F\) and the boundary data. As a byproduct, we obtain parabolic \(C^{1,\mathrm{Log\text{-}Lip}}\) regularity in the critical borderline case where the source term belongs to BMO space.

We study coarse topology of nodal sets of linear combinations of eigenfunctions of sub-Laplacians. More precisely, we prove coarse versions of Courant's and Bézout's theorems for linear combinations of eigenfunctions of sub-Laplacians on compact nilmanifolds obtained as quotients of stratified groups. We conjecture the extensions of these results to general closed equiregular sub-Riemannian manifolds and outline a programme for proving them. The method we use combines topological persistence and anisotropic Sobolev theory of Hörmander vector fields, generalizing the ideas which have recently been implemented in the Riemannian case.

We prove local smoothing estimates for wave equations with $C^{r}$ coefficients, relying on bilinear restriction estimates associated with rough wave propagation. In two and three dimensions, for $C^{1,1}$ coefficients we recover the sharp $L^p$-$L^q$ local smoothing bounds previously established for smooth coefficients. These are implied by stronger $L^{p}$-$L^{q}$ decoupling inequalities with endpoint derivative loss. As a corollary, we obtain non-trivial $L^{p}$-$L^{p}$ local smoothing estimates as well.

We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_{\Omega} g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(t)\,dt < \infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.

We study the conformally invariant exponential biharmonic Choquard equation in \(\mathbb R^{4}\). Our principal result is the complete spectral resolution of the linearized operator at its conformal bubbles. After stereographic projection, the operator becomes a bounded zeroth-order perturbation of the Paneitz operator, with a compact Riesz component on \(L^{2}(\mathbb S^{4})\). We justify the weak conformal transfer, remove every pole-supported distributional defect, and compute all eigenvalues. The Morse index is one. The kernel is the five-dimensional conformal space. All higher modes satisfy a uniform coercivity estimate. The transverse Hessian of the Adams–Choquard deficit is the same linearized operator. Hence the complete spectrum gives sharp local stability with respect to the Paneitz distance from the conformal extremal manifold. The optimal asymptotic constant is \[ \gamma_{\alpha} =\frac{160-12\alpha-\alpha^{2}}{40(10-\alpha)}, \] and the limiting quotient is minimized precisely by the second spherical harmonics. As a nonlinear preparation, we also prove that every normal distributional finite-mass solution satisfies the hypotheses of Niu's classification theorem and is therefore an explicit translation–dilation bubble.

This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.

We show that for every closed, smooth manifold $(M,g)$ of dimension $d$, there exists $c(g)$ such that any nodal domain $\Omega_\lambda$ of a Laplace eigenfunction with eigenvalue $\lambda$ contains a geodesic ball of radius at least $c(g) \lambda^{-1/2} \log\log(\lambda)^{-1/2}$ if $d=3$ and $c(g) \lambda^{-1/2} \log(\lambda)^{-\frac{d-3}{2}}$ if $d >3$. This ball is centered at any point at which the eigenfunction attains its maximum in absolute value within the nodal domain. Furthermore, we show that for any $d \geq 3$, there exist sequences of $\lambda$-nodal domains on $\mathbb{T}^d$ whose inner radius is of order $o(\lambda^{-1/2})$.

We construct multiple solutions of the generalized self-dual Abelian Chern-Simons-Higgs equation in a two dimensional flat torus by the topological degree method.

We show that Ekeland-type variational principles can be derived from steepest descent curves in metric spaces. From this, we relate the existence of the Calabi flow to the validity of the Ekeland variational principle for the $K$-energy on the space of Kähler potentials.

We study optimal data-driven solutions for the stationary diffusive population growth model $-\Delta u = r u$ in a bounded domain $\Omega\subset\mathbb R^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u,r)\in H^1(\Omega)\times L^\infty(\Omega)$ that fits a given data set in an optimal way measured by a cost functional $I$ and an additional penalty term. We characterize the relaxed cost functional sc$^- I$ by showing that its density is given as the partial lower convex envelope with respect to the variable $r$, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-\Delta u = \varrho(x,u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x,u)\mapsto \varrho(x,u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $\Gamma$-convergence of the associated cost functionals.

It is well-known the continuity properties of solutions to the wave equation depend strongly on both the initial data and the boundary conditions, particularly across the characteristic lines, where information propagates. Moreover, the continuity properties of solutions to the wave equation depend strongly on the type of boundary condition imposed and on the interaction of the characteristics with the boundary. Across the characteristic lines, singularities and discontinuities propagate according to the geometry of the problem, while the boundary conditions determine how these singularities are re?ected or transmitted. In this study, we will examine the behavior of solutions of the wave equation, corresponding to different boundary conditions, on its characteristic line, using the formulas obtained as a result of d'Alembert's formula, for all the three, Dirichlet, Neumann and Robin, types of boundary conditions.

We prove the second commutation lemma for the Lebesgue spaces $L^p({\bf R}^d)$, $1 < p < \infty$, on the whole unbounded domain, extending the $L^2$ theory of Tartar (1990) to the Banach-space framework of H-distributions. Unlike the $L^2$ setting, where the Plancherel isometry and Hilbert-space compactness are available, the global $L^p$ setting has neither. We control the non-local tail through Calderón–Zygmund kernel estimates, an explicit Taylor-remainder identity, and spatial truncation, obtaining order-$(-\epsilon)$ smoothing of the remainder into the Besov scale. Pairing weakly convergent sequences with their canonical Nemyckij duals, we then use the lemma to derive $L^p$ transport equations. For first-order scalar equations we establish the phase-space bicharacteristic (Vlasov) flow of the associated H-distribution when $p \ge 2$; for the quasilinear $p$-wave system we lift the local energy identity to the microlocal level, obtaining a Poynting-flux transport of microlocal energy in the linear core $p = 2$ and isolating the structural obstruction to its closure when $p \neq 2$. These results provide functional-analytic tools for tracking the propagation of singularities in degenerate nonlinear and non-local partial differential equations.

Accurate prediction of the nonlinear radiation thermal transfer in composite structures with temperature-dependent properties is significant in high-temperature applications of the materials. This study establishes a high-accuracy multi-scale computational model incorporating novel high-order correction terms for the high-fidelity simulation of nonlinear thermal radiation in composite structures, enabling local balance preserving of heat quantity. Moreover, an explicit convergence rate is also derived for the resulting high-order multi-scale solutions. Furthermore, an efficient multi-scale algorithm consisting of off-line and on-line computation stages is developed for high-accuracy simulation of nonlinear thermal radiation behavior in composite structures, and corresponding convergence analysis is also obtained. Two- and three-dimensional numerical examples are presented to validate the competitive advantages of the proposed multi-scale approach, not only exceptional numerical accuracy, but also reduced computational cost in both storage requirements and computational time.

We prove a conjecture of Datar-Mete-Song \cite{DMS} characterizing $J$-slope semi-stability by the minimal $J$-slope. More precisely, for a semi-stable pair of Kähler classes $(\alpha,\beta)$ on a compact Kähler manifold $X$, every big and nef birational test class has slope at least the topological $J$-slope, whereas an unstable pair admits a test class with strictly smaller slope. We also introduce the $J$-null locus of a semi-stable pair and prove that it is an analytic subset of $X$ if $X$ is a compact Kähler surface or a compact toric Kähler manifold. In the toric invariant case, we show that Murakami's \cite{Murakami} weak solution to the $J$-equation is smooth and Kähler on the dense big torus $(\mathbb{C}^*)^n$ of $X$.

We establish a sharp integral nonexistence criterion for the Lane–Emden system of inequalities \[ -\Delta u\ge v^p,\qquad -\Delta v\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty, \] then every nonnegative solution $(u,v)$ satisfies $u\equiv v\equiv0$. The proof combines flow decomposition of the finite Green current with nonlinear testing. In the symmetric case $p=q>1$, the Liouville problem reduces, via the sum $u+v$, to the scalar criterion \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.

For time-fractional gradient flows, natural dissipation statements are often integrated or memory-modified rather than pointwise differential inequalities for the original energy. We construct a smooth, compactly supported, globally semiconvex energy and an absolutely continuous solution on a finite time interval of a time-fractional gradient flow along which the original energy is strictly increasing on an explicit subinterval. The construction incorporates the initial layer directly into the trajectory. After transformation, the curve and its fractional derivative are polynomial, the curve is a regular immersion, and the Hessian remains bounded below up to the anchor.

We study the Wasserstein gradient flow of the squared Maximum Mean Discrepancy (MMD) generated by the nonsmooth energy kernels $K(z)=-|z|^q$, $0<q<2$. In dimensions $d\ge2$, the corresponding energies are not displacement semiconvex, so standard Wasserstein-gradient-flow theory does not apply. When $d+q-2>0$, we prove global well-posedness on $\mathbb{R}^d$ for probability densities in subcritical $L^p$ spaces, with targets in the same integrability class and with finite moments. We also include the one-dimensional Coulomb endpoint $d=q=1$. For the associated $N$-particle system, we prove global noncollision and fixed-$N$ convergence to the collision-free critical set, a particle-to-continuum criticality principle, and a modulated-energy mean-field estimate that yields convergence of the particle dynamics to the continuum flow as $N\to\infty$ on every finite time interval. We also construct collision-free saddle equilibria, showing that deterministic particle trajectories need not approach global empirical minimizers. For $1\le q<2$, every continuum solution in our class has a narrowly relatively compact orbit, every $\omega$-limit point is Lagrangian critical, and the orbit approaches the Lagrangian critical set. For $0<q<1$, the same conclusions hold under uniform-in-time moment and subcritical $L^p$ bounds. We prove that an absolutely continuous Lagrangian critical point equals the target when the source and target have finite moments of order $q$, except when $0<q<1$ and $d\in\{1,3\}$. Under the preceding uniform bounds, rigidity gives convergence of the continuum flow to the target throughout the rigid part of the well-posedness range. Finally, we show that no initial-data-independent multiplicative MMD decay modulus exists on $\mathbb{R}^d$, and that global Polyak–Łojasiewicz inequalities fail in several whole-space and periodic Riesz/Coulomb regimes.

The present work presents a real Paley-Wiener theorem within the space of multi-anisotropic ultradifferentiable functions. We also show new properties of this space.

In this paper, we study a class of nonlinear Schrödinger equations (NLS) in a unified way. This class includes two important examples: the intermediate NLS (INLS) and the Calogero–Moser derivative NLS (CM-DNLS). Our main results are twofold. First, we prove small-data global well-posedness in $L^2(\mathbb{R})$ for a broad class of equations. This includes both focusing and defocusing CM-DNLS and the INLS for arbitrary choices of its parameters. Second, we prove small-data scattering in $L^2(\mathbb{R})$ under an additional assumption. This result covers both focusing and defocusing CM-DNLS and the INLS for specific choices of its parameters. Both the formulation of the problem, including the notion of solution, and the proofs rely crucially on a linear theory for Schrödinger equations with rough time-dependent potentials. This theory is also of independent interest, since we allow potentials so rough that the standard Duhamel formulation may not make sense. Our approach is perturbative and is built on the bilinear Strichartz estimate proved by Ozawa and Tsutsumi in 1998. In particular, our argument does not rely on integrability.

In this paper, we introduce the lower and upper decay indicators and the associated decay character on the Heisenberg group. These notions are used to derive decay estimates for the linear fractional diffusion equation and to characterize the decay of initial data in several function spaces. We then study the Cauchy problem $$ \partial_t^2u+\left(-\Delta{\mathrm H}\right)^{\delta_1}u+\left(-\Delta_{\mathrm H}\right)^{\delta_2}\partial_tu=0, \quad \delta_1\in\left[0,\frac{\delta_2}{2}\right], $$ and establish decay estimates for solutions and their derivatives in homogeneous fractional Sobolev spaces in terms of the decay characters of the initial data. These results recover known estimates and extend them to new classes of data. We also investigate the corresponding semilinear problem with nonlinearity $|u|^p$. Global existence and decay are proved for $$ p>1+\frac{2\omega\delta_1}{Q-2\omega\delta_2}, \quad \omega=\frac{Q}{Q+\min{r_{\mathrm H}(u_0),r_{\mathrm H}(u_1)-2\delta_2}+2\delta_2}, $$ together with the corresponding critical case. Finally, by constructing test functions adapted to the nonlocal fractional sub-Laplacians, we establish blow-up results and identify the critical exponent $$ p=1+\frac{2m\delta_1}{Q+m\gamma-2m\delta_2} $$ for initial data in $\dot H_m^{-\gamma}(\mathbf H_n)$, where $m\in(1,2]$ and $\gamma\in\left[0,Q-\frac{Q}{m}\right)$.

We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0) > \frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.

In this paper, the logarithmic $p$-Laplacian operator $\log (-\Delta_{\mathbb H ^n})_p$ on the hyperbolic space $\mathbb H^n$, with $n\geq 2$, is introduced. We prove that if $f$ is a locally Lipschitz function of exponent $\alpha \in (0,1)$ with compact support in $\mathbb H^n$, then, for a suitable constant $A_{n,p}>0$, $$ \lim_{s\rightarrow 0^+}(-\Delta_{\mathbb H ^n})_p^sf(x)=A_{n,p}|f(x)|^{p-2}f(x),\quad x\in \mathbb H^n, $$ where $(-\Delta_{\mathbb H ^n})_p^s$ denotes the $s$-fractional $p$-Laplacian on $\mathbb H^n$. We establish a pointwise integral representation for the operator $\log (-\Delta_{\mathbb H ^n})_p=\frac{d}{ds}(-\Delta_{\mathbb H^n})_p^s\,_{|s=0}$. Furthermore, we show that $\log (-\Delta_{\mathbb H ^n})_p$ can be realized as the solution of a suitable extension problem and provide an extension theorem that yields the operator $\log (-\Delta)_p$ in $\mathbb R^n$. To the best of our knowledge, this property has not been established for the Euclidean logarithmic $p$-Laplacian $\log (-\Delta)_p$.

We establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation. Since the curvature matrix depends on both $Du$ and $D^2u$, a phase subsolution alone does not provide the full linearized separation needed for the mixed derivative estimate. We introduce a mixed recession compatibility condition imposed only on doubly degenerate level jets. It yields a uniform mixed derivative bound and is sharp within the class of fixed smooth zero-order barriers considered here. For the double-normal derivative, an exact complex Schur-complement identity gives \[ u_{\nu\nu}=\beta+\alpha\cot\delta, \qquad 1\leq\alpha\leq C, \qquad |\beta|\leq C, \] where $\alpha$ and $\beta$ are explicit Schur-complement coefficients, $\delta$ is the actual boundary limiting-phase gap, and $C$ depends only on uniform bounds for the gradient and the mixed boundary derivatives. Thus curvature blows up if and only if this gap collapses, with optimal rate $\delta^{-1}$. Smooth radial solutions attain the rate, while a rank-loss model shows that a strict lower subsolution need not force strict convexity.

We consider the two-dimensional incompressible Euler equation on $\mathbb T^2$ with Gaussian random initial data having independent Fourier coefficients. For every $\sigma>0$, we prove that such a Gaussian measure on $H^\sigma(\mathbb T^2)$ is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.

We study the three-dimensional incompressible Boussinesq equations on a bounded domain with smooth boundary and Navier boundary conditions. We construct global weak solutions by a Galerkin approximation and establish the associated Leray–Hopf energy inequalities. For nonnegative boundary friction, the total energy decays exponentially when the friction coefficient is positive on a boundary subset of positive surface measure. In the frictionless case, the scalar field and the velocity component orthogonal to the rigid-motion kernel decay exponentially; when the kernel is trivial, this is exponential decay of the total energy. When the scalar initial datum vanishes, this also proves exponential decay for the Navier–Stokes system on solids of revolution for every friction coefficient $\alpha\in L^\infty(\partial\Omega)$ such that $\alpha\ge0$ almost everywhere and $\alpha\not\equiv0$, resolving the corresponding case left open in \cite{Kelliher2025}. The proof uses a weighted Korn–Poincaré inequality and a two-time Gronwall-type inequality with an exponentially decaying forcing term.

In this paper, we study the semilinear wave equations $$ u_{tt}-\Delta u+b(t)u_t+m^2(t)u=|u|^p, \quad t \geq 0, \quad x\in\mathbb{R}^n $$ with effective time-dependent damping and a time-dependent mass dominated by the damping. D'Abbicco, Girardi and Reissig established global small-data existence in supercritical ranges and identified the scale $$ p_{\beta,\eta}(n)=1+\frac{2\eta}{n+2\eta\beta} $$ for initial data in $(L^\eta(\mathbb R^n)\cap H^1(\mathbb R^n))\times(L^\eta(\mathbb R^n)\cap L^2(\mathbb R^n))$ with $1\leq\eta<2$, where $\beta$ is the lower mass index associated with the damping-mass pair. To support the expected sharpness of this scale, they also established an analogous subcritical nonexistence result for the corresponding diffusion equation with nonnegative initial data in $L^\eta(\mathbb{R}^n)$, leaving the wave-equation counterpart with effective damping and time-dependent mass open. We address this problem for $\eta=1$ under an intrinsic accumulated-mass balance and a Liouville nonoscillation condition. By constructing a positive slow adjoint mode, we prove nonexistence of global weak solutions for $$ 1<p<p_{\beta,1}(n)=1+\frac{2}{n+2\beta}, $$ and also treat the critical case $p=p_{\beta,1}(n)$ under an Osgood divergence condition. Conditional lifespan upper bounds and explicit admissible coefficient families are also given.

We consider the defocusing energy-supercritical nonlinear Schrödinger equation $i\partial_{t}u+\Delta u=|u|^p u$ in dimensions $d\ge5$. Killip-Visan [Comm. Partial Differential Equations, 2010] and Li-Li [Siam J. Math. Anal., 2022] proved that for $s_c:=\frac{d}{2}-\frac{2}{p}>1$, any solution that remains bounded in the critical Sobolev space $\dot H_x^{s_c}(\mathbb{R} ^d)$ must be global and scatter. In dimensions \(d \ge 8\), their results required either that \(p\) be even or that \(s_c < \frac{d+2-\sqrt{(d-2)^2-16}}{4}\). In this paper, we improve the upper bound on \(s_c\) to \(s_c<1+p\) by establishing some new nonlinear estimates. This allows us to cover all cases in which \(p\) lies in the local existence range.

The goal of this article is to derive observability properties for wave and plate equations with rough coefficients in the principal part, including the case of interfaces, through a regularization process. Specifically, we demonstrate that if a singular coefficient can be approximated by a sequence of smooth coefficients corresponding to uniformly observable systems, then the observability inequalities can be passed to the limit. This allows us to derive observability properties for the limit system. While this strategy is natural, we show that it can be used to establish new observability results in several settings, particularly in the presence of interfaces meeting the boundary, or multiple interfaces intersecting at a point, under a suitable multiplier-type condition that imposes sign constraints on the jumps of the coefficients at the interfaces. A key advantage of this approach is that it avoids the need for a detailed analysis of the regularity of solutions near the sets where the coefficients are singular.

We address the prescribed Lorentzian mean curvature problem over a convex bounded domain $\Omega$ of $\mathbb R^m$ with bounded right-hand side and homogeneous capillary boundary condition. We prove that the problem has a unique $W^{2,2}$-regular weak solution $u$ with zero mean and that $|Du| \leq 1 - \theta$ for some $\theta\in(0,1)$ only depending on the data. Such $u$ is also the unique maximizer of an associated functional. A key step in proving that the maximizer is a weak solution consists in showing that it has no light segments, i.e. segments along which $|Du| = 1$. This holds for arbitrary bounded capillary boundary data and can thus be an interesting result on its own.

This work is concerned with weighted Geometric Rigidity Estimates and Korn's first inequalities in bulk and thin domains. We consider weights, that are a nonnegative power of the distance function to part of the boundary of the domain. For the case of thin domains, the distance is taken to be from the thin face of the domain boundary, and the constants depend on the domain thickness and when the mid-surface contains a flat region, the constants are proven to have optimal scaling as the thickness approaches zero. We employed some covering techniques utilized by Acosta, Cejas, and Duran in [\ref{bib:Aco.Cej.Dur.}] to prove a weighted Poincaré inequality, and later by Conti and Zwicknagl in [\ref{bib:Con.Zwi.}] to prove weighted Poincaré and classical Geometric Rigidity inequalities in Lipschitz domains. However, because the distance is taken to be only from part of the boundary, the covering parts become more delicate, especially for thin domains and the analysis becomes non-straightforward.