Papers

We construct planar simply-connected domains whose torsion function has an arbitrary prescribed number of local maxima while the distance to the boundary has only one local maximum inside the domain. This disproves a conjecture of Magnanini asserting that the number of local maxima of the torsion function is bounded above by the number of local maxima of the distance function to the boundary. We also answer a question of Steinerberger concerning the eccentricity of level sets near the point where the torsion function attains its maximum in simply-connected domains.

We study a system in which a fluid occupying a bounded simply connected region in $\mathbb{R}^{2}$ is surrounded by another fluid with sharp boundary. They are incompressible Brinkman flows of the same viscosity saturating a porous medium with constant permeability. They interact via surface tension on their interface. We assume that the velocity has no jump across the interface and decays at infinity. We establish the asymptotic stability of the circular interface, which is a steady-state solution to our system. The technical threshold for the size of the initial perturbation for asymptotic stability can be explicitly calculated. We further show that the initial perturbation decays exponentially. We prove the existence, uniqueness, and continuous dependence on initial data of highly regular solutions by containing the perturbation variable in a Wiener-type algebra with a time-activated exponential weight, which allows for analytic solutions with much coarser initial data. The solution is contained in the Wiener-type algebra with the critical scaling exponent for our problem.

We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain $\Omega\subset\mathbb{R}^d$ by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal $\mu_x\propto e^{-U}$ satisfies a Poincaré inequality on $\Omega$ with constant $m>0$, $\nabla^2U\succeq-K\,\mathrm{Id}$ and $\Omega$ is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in $L^2$, with an explicit rate that scales like $\sqrt m$, which is optimal when $U$ is convex. Since the normally reflected overdamped dynamics converges exactly at rate $m$, this establishes a square-root acceleration for constrained sampling in the small-gap regime when $m$ is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan–Li–Lu. The proof adapts the modified $L^2$ hypocoercivity method of Dolbeault–Mouhot–Schmeiser with the gap-shifted corrector of Fan–Li–Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of $\partial\Omega$ and is nonnegative for convex $\Omega$. Finally, we extend our results to the setting where the domain $\Omega$ is non-convex. We obtain an explicit contraction rate that depends on the domain.

This paper studies generalized Hamiltonian gradient flows for contact-type Hamilton–Jacobi equations, adopting the variational framework of Herglotz's principle and its fundamental solution \(h_L(t,x,y,u)\). Two main results are presented. First, precise first-order sensitivity relations are derived, linking derivatives of \(h_L\) to dual arcs of minimizing trajectories. Second, quantitative second-order estimates show that \(h_L\) is locally semiconcave and, over short time, semiconvex, indeed uniformly convex in certain variables. These regularity properties follow from a detailed variational analysis of minimizing trajectories. The work establishes a foundation for intrinsic methods in analyzing singularity propagation and generalized gradient flows in contact context, with implications for weak KAM theory, optimal transport, and regularity of viscosity solutions.

This paper provides a complete characterization of the sign of the propagation speed in the symmetric two-species Lotka-Volterra competition-diffusion model under strong competition. The system admits a unique bistable travelling front and the sign of its speed determines which of the two species invades the other. We prove that for every strong competition intensity greater than $1$ and every diffusion ratio $d\neq1$, the travelling front propagates so as to expand the territory occupied by the faster-diffusing species. The front has a zero speed exactly when $d=1$. We also establish the smooth dependence of the wave speed and the travelling front on the model parameters. The main step in the sign characterization is to prove that a monotone standing front cannot exist when the diffusion rates are different. Combined with continuity of the wave speed with respect to the parameters, the species-exchange symmetry of the system, and an explicit travelling front at a particular parameter value, we obtain the sign of the propagation speed throughout the entire parameter region. This establishes the “Unity is not strength” theorem, which was previously known only in restricted parameter regimes.

Though the sharp $L^{2}$-Caffarelli–Kohn–Nirenberg (CKN) inequalities have been extensively studied in the entire Euclidean spaces, the corresponding problem on domains whose boundary contains the origin remains largely unexplored. We investigate the sharp $L^{2}$-CKN inequalities on half-spaces and orthants $\mathbb R^{n}_{k,+}$ by computing explicitly the optimal constants, determining all possible extremal functions, and establishing exact identities for the deficits. Since the singular weights $|x|^{-2b}$ rule out the lifting argument that is available for the simpler Heisenberg Uncertainty Principle, we develop an approach based on the transformations $u(x)=|x|^{m}v(x)$ for an appropriately chosen $m$ combined with spherical harmonic decompositions and weighted identities. Moreover, we establish weighted Poincaré inequalities associated with measures of the form \[ e^{-\delta|x|^{\tau}}|x|^{\beta}\bigl(\prod_{i=n-k+1}^{n}x_i^{2}\bigr)\,dx, \] together with their sharp constants, extremizers and stability estimates, which substantially extend those of the classical Gaussian Poincaré inequality. On the full orthant, the linear modes cease to be admissible competitors, since all odd spherical harmonics are annihilated by the lifting; the first non-radial mode is then of degree two, and both the sharp constant and the manifold of optimizers change accordingly. Finally, we establish several stability estimates, and second-order stability estimates, of the CKN inequalities on the half-spaces and orthants throughout the full parameter range.

We study the quantum drift-diffusion, or Derrida-Lebowitz-Speer-Spohn (DLSS), equation for a nonnegative density $\varrho$ on a bounded convex domain with Neumann boundary conditions, in the square-root variable $u=\sqrt\varrho$. We show that the DLSS operator, defined and monotone on smooth strictly positive functions, admits a unique maximal monotone extension in $L^2(\Omega)$, explicitly given by the minimal (defect-free) operator plus the normal cone of the positivity constraint. The generated semigroup, which contracts the Hellinger distance between the densities, thus yields a canonical solution - existing, unique, and stable for every nonnegative $L^2$ initial datum and in every space dimension - independent of any approximation scheme: it is in fact the unique contraction semigroup extending the classical evolutions that emanate from smooth, uniformly positive data. The implicit Euler scheme converges to it, and $\sqrt u\in L^2_{\rm loc}(H^2)$ along the flow. When the datum belongs to the domain of the operator, the solution is strong and satisfies the equation pointwise, with no reaction term created on the vacuum $\{u=0\}$. We characterize the trajectories in several equivalent ways - as Bénilan integral solutions and through one-sided weak formulations - prove the maximality of the operator also in the $H^2$-$H^{-2}$ duality and, in dimension $d\le3$, identify the flow with the weak solutions in the uniqueness class of Fischer. A second-order estimate of independent interest underlies the construction: on a convex domain with Neumann conditions the dissipation $\int_\Omega(\Delta u)^2/u\,\mathrm{d} x$ is finite exactly when $\sqrt u\in H^2(\Omega)$, and it then controls the full Hessian of $\sqrt u$, in every dimension.

The aim of the present paper is to develop a spectral perturbation theory from a higher perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, our first main result establishes a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which only depends on the “data”, i.e. the limit eigenspace and the magnitude of the perturbation, through the resolution of a minimization problem. Remarkably, we make no assumptions on the perturbed eigenelements (besides, naturally, the spectral stability). Moreover, we cover both the cases of simple and multiple limit eigenvalues in full generality. In the second part, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights (also motivated by optimization in spectral geometry); secondly, we investigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Riemannian manifold changes when a second small manifold is glued on a small portion of it.

Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularity mechanism. First, a local Pogorelov model is chosen to solve the exact equation $\det D^2u=1$ a.e. It is convex, belongs to $C^{1,1-2/n}\cap W^{2,n}_{\loc}$, and its Hessian is positive definite off a line. Nevertheless, its gradient collapses that line to one point. In fact the gradient is neither open nor discrete, its branch set is exactly the collapsed line, and its Jacobian equals one a.e. We then develop a $k$-dimensional version of the construction. If $1\leq k<n/2$, there are convex potentials with a $k$-dimensional flat contact set, uniformly positive and bounded Hessian determinant, and exact critical exponent \[ p_{n,k}=\frac{n(n-k)}{2k}. \] Their Hessians belong to $L^p_{\loc}$ precisely for $p<p_{n,k}$, lie in the weak endpoint space $L^{p_{n,k},\infty}_{\loc}$, and fail to belong to any finite-index Lorentz endpoint. This matches the critical minimum-set theorem of Collins and Mooney. At the regularity required in the question, the construction produces branch sets of every integer dimension $k<n/3$. We also give a topological reformulation of the problem. In the convex branch, gradient fibers are exactly contact sets with supporting affine functions, and openness and discreteness are equivalent to strict convexity. For a general gradient under the hypotheses above, critical Sobolev mapping theory already supplies continuity and sense preservation. The question asks whether the monotone factor in the Eilenberg–Whyburn monotone–light factorization is trivial.

We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) 58 (2025), no.~4, 943–970], which treats $m$-fold symmetry solutions with $m\geq4$. We prove that every nonconstant $W^{1,p}$ solution satisfies $\|g_\theta(t)\|_{L^p}\to\infty$ as $t\to\pm\infty$ for $1<p\leq\infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D_\theta g_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{\alpha,r}$ for $\alpha r<1$. For such data, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit. This structural assumption on $D_\theta g_0$ is automatic for $C^1$ data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. 10 (2023), no.~1, 1–100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for $m\geq4$, they cover the full well-posed scale-invariant range $m\geq3$.

We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.

We consider a degenerate chemotaxis system of the form \begin{align}\label{star}\tag{$\star$} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\nabla\cdot\big(D(u)\nabla u-uS(u)\nabla v\big)+f(u,v),\\ &v_t=\Delta v+g(u,v),\end{array}\right. \end{align} in a bounded domain $\Omega\subset\mathbb{R}^{N}$ with smooth boundary subjected to no-flux and homogeneous Neumann boundary conditions. Herein, the diffusion coefficient $D\in C^0([0,\infty))\cap C^1((0,\infty))$ is assumed to satisfy $D(0)=0$ and $D'(s)\geq 0$ on $(0,\infty)$, and there are $s_0\in(0,1]$ and $d>0$ such that $D(s)\geq ds^{m-1}$ on $[0,s_0]$ and that \begin{align*} s D'(s)\leq C_D D(s)\quad\text{for }s\in[0,s_0]. \end{align*} The sensitivity function $S\in C^2([0,\infty))$ and the source term $f\in C^{1}([0,\infty)\times[0,\infty))$ in the first equation are supposed to be nonnegative. The source term $g\in C^{1}([0,\infty)\times[0,\infty))$ of the second equation can in fact be negative. Prototypical choices for $g$ are $g(u,v)=-uv$ and $g(u,v)=-v+u$. We show under suitable assumptions on weak solutions to \eqref{star} on $\Omega\times(0,T_0)$, that whenever the smoothly bounded domain $\omega\subset\mathbb{R}^N$ and $T\in(0,T_0)$ are such that \begin{align*} \overline{\omega}\subseteq \Omega,\qquad u_0>0\ \text{ in }\ \overline{\omega},\qquad\text{ and }\qquad u>0\ \text{ on }\ \partial\omega\times(0,T), \end{align*} then \begin{align*} u>0\quad\text{in }\ \overline{\omega}\times[0,T). \end{align*} In particular, any dead cores that appear during the evolution must have developed from regions that were already part of the initial zero set.

We establish quantitative mean-field convergence and propagation of chaos for repulsive Coulomb gradient flows at the bounded-density regularity of the limiting equation. The argument couples the dissipative modulated-energy identity with the normalized quadratic transport cost of the full $N$-particle law. The remaining negative mean-square force-error term is used through an exact completion of squares after mollification: the non-Lipschitz remainder is absorbed by this negative term, while a sharp first-order commutator estimate is applied to the mollified Lipschitz field. For the Coulomb equation, the sharp $L^\infty$ decay gives the density envelope $m(t)=\|\rho_0\|_{L^\infty}/(1+t\|\rho_0\|_{L^\infty})$. A density-adapted transport weight and mollification scale $m(t)^{-1/d}$ yield an Osgood comparison. Thus, for every $d\ge2$ and $\rho_0\in\mathcal P_2(\mathbb R^d)\cap L^\infty(\mathbb R^d)$, we obtain quantitative comparison with the global bounded-density Coulomb solution on every prescribed finite interval. For tensorized initial data, the normalized squared Wasserstein distance of the full $N$-particle law, the expected modulated energy, and the time-integrated mean-square force error are bounded by $N^{-2\gamma_{T,d}/d}$ for $d\ge3$ and $((1+\log N)/N)^{\gamma_{T,2}}$ for $d=2$, where $\gamma_{T,d}=(1+T\|\rho_0\|_{L^\infty})^{-c_d}$. For $d-2<s<d$, we also prove Riesz weak–strong stability for prescribed reference solutions in $L^\infty(0,T;B^{s-d+2}_{\infty,q})$, with Gronwall, Bihari, and Osgood comparisons according to $q$, together with uniqueness in the stated Besov class. Finally, an outlier construction separates modulated-energy convergence and Kac chaos from normalized Wasserstein convergence of the full $N$-particle law.

We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \frac{\alpha}{\varepsilon} \beta\left(\frac{u_\varepsilon}{\varepsilon}\right) \qquad\text{in }B_1\subset\mathbb R^n, $$ where $F$ is uniformly elliptic and $\beta\in C_c (-1,1)$ is nonnegative. We prove the scale-sharp estimate $$ \|\nabla u_\varepsilon\|_{L^\infty(B_{1/2})} \leq C\left( \|u_\varepsilon\|_{L^\infty(B_1)}+\sqrt\alpha \right), $$ with $C$ depending only on the dimension, the ellipticity constants, and $\beta$, and independent of $\varepsilon$ and $\alpha$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\varepsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva–Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.

We develop a new convex-integration scheme, inspired by \cite{BCK26}, for the inviscid surface quasi-geostrophic equation on the two-dimensional torus. For the explicit, nonoptimized exponent $\bar p=\frac{4}{3}+10^{-5},$ we prove a flexibility theorem for weak solutions in the standard momentum formulation with active scalar \[ \theta\in C([0,1];L^{\bar p}(\mathbb T^2)). \] More precisely, any two prescribed mean-zero states in $L^{\bar p}(\mathbb T^2)$ can be approximated at the initial and final times by such a solution. The perturbations are constructed from localized, concentrated traveling SQG profiles whose centers move along rational directions and whose radii depend on the Reynolds stress. Time averages of auxiliary sources along these trajectories reconstruct the preceding-stage stress, while a two-dimensional bilinear null-form estimate compensates for the derivative loss caused by the nonlocal constitutive law. Exploiting the time-locality of the iteration, we also obtain a dense subset of the mean-zero space $L^{\bar p}(\mathbb T^2)$ such that every initial datum in this subset admits at least two distinct momentum weak solutions. Thus, the construction establishes both flexibility and nonuniqueness beyond the concentration-critical exponent $p=4/3$.

Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $\Sigma^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminus\Sigma$ satisfying $\lambda(g^{-1}A_g)\in\Gk$ and $\sigma_k(g^{-1}A_g)=\kappa>0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

We study the dynamic behavior of the free boundary of the Hele-Shaw flow with a nonnegative source, in which the pressure satisfies $-\Delta p=c$ in $\{p>0\}$ and the free boundary moves with normal velocity $V=|\nabla p|$. For initial domains satisfying an interior ball condition, we prove that the patch $\{p>0\}$ expands at a locally positive rate, so that its hitting time is locally Lipschitz; this is optimal, since distinct portions of the patch may collide. We also show that the free boundary is locally either regular, or of collision type, or undergoing parabolic-scale hole closings of various dimensions: such a space-time characterization of singularities is new in Hele-Shaw flow. We further show that, near any time at which no collision occurs, the space-time free boundary is a $C^1$ hypersurface. This implies that at such times every free boundary point has a space-time normal: the space-time unit normal is parallel to the time direction precisely where the spatial free boundary has singular geometry. The results apply in particular to the tumor growth model with nutrients introduced by Perthame, Quirós, and Vázquez, but the analysis of singularities is new even for the classical injection problem.

We study the two-dimensional stationary incompressible Navier–Stokes equations on $\mathbb R^2$ with fractional dissipation $(-\Delta)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and constant pressure. This is a fractional counterpart of the planar finite-Dirichlet theorem of Gilbarg and Weinberger at $s=1$. The proof uses different arguments in three ranges. For $0<s<\frac13$, we combine an $\mathrm L^2$-estimate derived from the equation with a stream-function truncation argument. For $\frac13\leq s\leq\frac23$, we use a localized energy estimate whose boundary terms are supported on expanding annuli. For $\frac23<s<1$, we establish regularity and decay via a Lorentz-space bootstrap and then apply the maximum principle to the vorticity. We also treat the stationary damped Euler system at $s=0$ by combining the Bernoulli identity with a cut-off argument under an annular growth condition that includes $u\in \mathrm L^r(\mathbb R^2)$ for every $1\le r\le2$.

We prove a concavity inequality for the Hessian quotient operator $\sigma_k/\sigma_{k-1}$ using a change of basis for symmetric polynomials. This removes an additional structural concavity assumption previously imposed in the interior $C^2$ estimate for convex solutions of the $\sigma_k/\sigma_{k-1}$ equation. The method also yields several useful inequalities for elementary symmetric polynomials.

In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group $\mathbb H^n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}_{\mathbb H^n} u=|u|^{2^\star-2}u+f(\xi) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where $ \mathcal{L}_{\mathbb H^n} $ is the sub-Laplacian on $\mathbb H^n$, $2^\star=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\geq 1$, $S^{1,2}(\mathbb H^n)$ is the homogeneous Sobolev space on $\mathbb H^n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{1,2}(\mathbb H^n))'$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.

We consider a Choquard problem in a smooth bounded domain in the plane. We show the existence of solutions which concentrate at a finite number of points. Up to our knowledge, this is the first existence result for concentrating solutions for Choquard-type problems in planar domains.

Let $n\ge2$, $1<p<2$, $0<s<1$, and $sp>p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,\alpha}$ for some $\alpha=\alpha(n,p,s)>0$. This settles the open problem of interior gradient Hölder regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an affine-invariant improvement-of-flatness argument with a Liouville theorem for globally Lipschitz entire solutions. In the large-slope regime, the shifted Bregman energies converge to an anisotropic stable form of order $sp-p+2>1$. In the bounded-slope regime, the Liouville theorem follows from rigidity of extremal secants, a recurrent blow-down argument, and a directional Morrey-Kato estimate for the singular linearized kernel. An affine Campanato argument controls the variation of the best affine approximations across scales. These estimates yield a scale-invariant decay of the affine excess and hence the local $C^{1,\alpha}$ estimate.

In this article, we study the convergence rates of variational eigenvalues of the $p$-Laplacian with rapidly oscillating weights and potentials belonging in a suitable $L^q$ space. Our analysis covers both Dirichlet and Neumann boundary conditions, and we derive explicit estimates in terms of the eigenvalue index $k \in \mathbb{N}$ and the oscillation parameter $\varepsilon > 0$. These results are obtained through a detailed examination of certain oscillatory integrals and extend previously known results in the literature.

We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N = 6$. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $e^{-|t|}$. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension $N \geq 7$. The six-dimension case presents specific difficulties, as the ground state does not belong to $\dot H^{-1}$. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.

This paper investigates the global existence and non-negativity of weak solutions to an initial-boundary value problem for a one-dimensional fourth-order nonlinear degenerate parabolic equation. This model governs the convection phenomena in thin films driven by surface tension. Our analytical approach begins with the formulation of a regularized problem and a corresponding Galerkin approximating scheme. We first establish the existence of solutions to the approximate problem. Subsequently, by constructing specialized energy and entropy functionals, we derive uniform a priori estimates for the approximating solutions. Leveraging the Aubin-Lions compactness lemma, we pass to the limit and establish the non-negativity of the limit function. Finally, we demonstrate that this limit is indeed a global weak solution to the original initial-boundary value problem.

We study resonance prediction in dispersive media, formulated as nonlinear spectral problems for volume integral operators. The main idea is to use asymptotic analysis not only as a baseline approximation, but also as a guide for constructing predictive correction models. We learn the residual between asymptotic and reference resonances using features suggested by the subwavelength expansion, including the logarithmic scales specific to two dimensions. The resulting corrections substantially improve single-resonator and dimer predictions, and symbolic regression produces compact formulas for the learned residual. The results show that asymptotic analysis can be used not only to approximate resonances, but also to design the feature space in which data-driven corrections become accurate, low-dimensional, and interpretable.

We prove a sharp lower bound for the first nonzero Neumann eigenvalue of geodesic triangles of given diameter in two-dimensional space forms. The bound is given by the first positive radial Neumann eigenvalue of an one dimensional model; it is approached by degenerating isosceles triangles. When \(K>0\) and \(D=\pi/(2\sqrt K)\), equality is attained precisely by birectangular triangles. We also prove a hot-spots theorem for non-acute spherical triangles of diameter at most \(\pi/2\), and establish antisymmetry and eigenvalue monotonicity for isosceles spherical triangles of diameter \(\pi/2\).

We study an inverse source problem for a linear non-autonomous parabolic equation with additive source term $f(t)+\zeta(t,x)$, where the unknown component depends only on time and is recovered from an integral observation of the solution. After reducing the problem to an equivalent linear inverse problem, we establish existence and uniqueness of the Tikhonov-regularized solution and derive a first-order optimality condition. We show that the forward operator admits a Volterra representation in time, yielding a weak-norm stability estimate in $H^{-1}(0,T)$ and, under an a priori $H^r(0,T)$ bound on the source, a conditional Hölder stability estimate in $L^2(0,T)$. Motivated by this Volterra structure, we introduce an adjoint-free integral feedback method that reconstructs the source using only forward solves. We analyze the feedback iteration by establishing its well-definedness and fixed-point properties, convergence for exact data, and finite-iteration stability with respect to noisy data. We further show that, with an appropriate noise-dependent stopping rule, the method constitutes an iterative regularization scheme. Numerical experiments for smooth and piecewise constant sources, supplemented by temporal regularization and automatic parameter selection, demonstrate accurate and stable reconstructions in the presence of noise.

We study weighted $L^2$ Hessian estimates for parabolic terminal equations under the marginals of a prescribed diffusion, allowing singular initial laws and different principal coefficients in the equation and the diffusion. Under coefficient regularity, uniform ellipticity and bounded reference drift, we characterize the limiting optimal constant uniformly over shrinking subintervals with zero terminal data. Its finiteness is equivalent to linear well-posedness in the weighted parabolic Sobolev space on the full interval. For initial laws satisfying weighted doubling and coercivity conditions, this constant is the maximum of contributions from heat evolutions of rescaled initial measures, constant weights and exponential mixtures. The phase contributions are bounded above in terms of the coefficients at time zero outside the initial support and at spatial infinity. Under radial reference structure, we obtain exact coefficient formulas invariant under bounded changes of reference drift. In the matching case, with reference covariance $a$, principal matrix $a/2$ and normalized Hessian $a^{1/2}D^2u\,a^{1/2}$, the constant is $2$ for every initial law with time weight $t^\alpha$, $\alpha\ge1/2$; point starts attain $2\sqrt2$ at $\alpha=0$. Whenever the limiting constant is finite, increasing damping makes the full-interval Hessian constant converge to it and the value and gradient gains tend to zero. As applications, we obtain nonlinear solvability and residual stability for small Hessian perturbations. For solutions and candidates with a common regularity bound, secant estimates give linear stability under a relative spectral gap, while covariance interpolation extends residual stability beyond it.

Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay type log-periodic solutions posed by del Pino–Musso–Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron–Ponce–Véron (2007). We construct a global continuum of positive log-periodic solutions containing the local bifurcation branch and prove that blow-up along this continuum occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scaling-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli–Gidas–Spruck (1989).