Papers

This paper surveys different topics where the theory of quantitative rectifiability plays a central role. First, it reviews the characterization of rectifiability in terms of square functions involving $\beta$ type coefficients and the $\varepsilon^2$ conjecture of Carleson. It also discusses the deep connections between rectifiability and the $L^2$ boundedness of Riesz transforms and their application to the Painlevé problem for Lipschitz harmonic functions. Finally, the paper explores recent major advances in connection with harmonic measure and the $L^p$ solvability of the Dirichlet, regularity, and Neumann problems for the Laplace equation in rough domains, emphasizing the key role of quantitative rectifiability in these developments.

We study regularity properties of minimizers for the one-phase Alt–Caffarelli problem associated with the fractional $p$-Laplacian in the range $p\geq2$. We consider minimizers of the fractional $p$-energy penalized by the measure of the positivity set, with prescribed nonnegative exterior datum. We prove existence of minimizers and derive their basic variational properties: minimizers are nonnegative and are weak subsolutions of the homogeneous fractional $p$-Laplace equation. The main regularity argument combines fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, and a Campanato-type iteration. This yields local Hölder continuity of minimizers and implies that they solve the homogeneous equation in their positivity set. Finally, we prove the optimal free boundary growth estimate, showing that the sharp order known in the linear fractional Bernoulli problem persists in the fractional $p$-Laplacian setting.

We construct stationary solutions of the three-dimensional incompressible Euler equations with helical symmetry and support in a neighborhood of a helix. The solutions are piecewise smooth and arise from a nonlinear overdetermined elliptic boundary value problem associated with a stream-function formulation. A distinguishing feature is that the vortex cross-sections are intrinsically anisotropic: after rescaling, the leading-order shape is elliptic rather than radial, and the boundary exhibits a nontrivial third Fourier mode reflecting helical effects absent in previous axisymmetric constructions. A key step in the proof is the analysis of a genuinely anisotropic overdetermined elliptic problem with prescribed Dirichlet and nonconstant Neumann conditions.

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $\delta$; the critical regime $\varepsilon = \delta$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (\delta)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier–Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (\delta^{1 + \iota})$ for some $\iota > 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

The main purpose of this paper is to study the global solvability for a general class of quasilinear shifted wave equations on hyperbolic spaces, for smooth initial data with small amplitude. In contrast to the case of Euclidean spaces, when the space dimension is three, we do not need to assume structural conditions like the null conditions to ensure global existence. To achieve this, we establish the energy and local energy estimates for perturbed wave operators on $\mathbb{R}\times \mathbb{H}^n$. These estimates allow time-dependent metric perturbations and require only suitable smallness together with polynomial decay in the radial variable $r$. As a byproduct, for semilinear problems with power-type nonlinearities and radial data, we also obtain global solutions with low-regularity. In particular, we prove an analog of the radial Glassey conjecture on hyperbolic space.

We study the inertial motion of incompressible continua within a geometric and variational framework, extending the classical Arnold-Ebin-Marsden theory from the group of volume-preserving diffeomorphisms of a fixed domain to configuration spaces of deformations with variable image. In the latter case, the lack of a group structure requires proving some results that are instead immediate in the classical setting. We show that the orientation-preserving deformations with suitable regularity constitute a Hilbert manifold and that volume-preserving deformations form a submanifold with tangent vectors that are mapped onto divergence-free vector fields by the Lagrangian-to-Eulerian-picture correspondence. In so doing, we also present the geometric structure corresponding to compressible continua. The kinetic energy of the continuum gives rise to both a Lagrangian action, from which the equations of inertial motion are deduced, and a metric, with geodesics that are identified precisely by inertial motions. While the inertial motion can coincide with a physical one only for incompressible perfect fluids, it can be used to provide a natural parametrization of the configuration manifold for generic continua. We obtain a general result of local-in-time existence of solutions for the geodesic flow equation and we present explicit examples showing that, for the same initial data, the geodesic followed by the continuum in the compressible case can leave the manifold of admissible deformations in finite time, while the corresponding incompressible geodesic exists for all times.

We variationally characterize the bottom of the spectrum of the buckling problem in an infinite cylinder $A\times\mathbb{R}^{n-d}$, where $A$ is an open bounded subset of $\mathbb{R}^d$, and compute it explicitly when $A$ is a ball.

We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu h(x,u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \] in a bounded domain $\Omega \subset \mathbb{R}^N$, where $\lambda, \mu$ are real parameters, and the nonlinearity $h$ is superlinear at $u = 0$. In particular, we study the combined effect of the parameters $\lambda,\,\mu$ on the multiplicity of solutions. In the general setting, following Szulkin's approach for nonsmooth functionals, we prove the existence, for $\lambda$ not belonging to the spectrum of the Dirichlet Laplacian and $\mu$ sufficiently large, of a global minimizing solution (with negative action level) and of a min-max solution (with positive action level). Moreover, we characterize the limiting profiles of these solutions as $\mu \to +\infty$. More precisely, when the global minimizer is positive, its limit profile is $\mathrm{dist}(\cdot,\partial\Omega)$, thus saturating, in the limit, the geometric constraint $|\nabla u|\le1$, while min-max solutions collapse uniformly to zero as $\mu\to+\infty$. A nonexistence criterion is also given for suitable values of $\lambda$ and $\mu$. Finally, when the domain $\Omega$ is a ball, using a shooting approach, we establish the existence of arbitrarily many nodal radial solutions for every $\lambda \ge 0$ and for $\mu$ sufficiently large.

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)\beta(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $\beta$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv \delta_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

We study the joint diffusive-mean field limit for a system of weakly interacting kinetic Langevin dynamics, extending the results of Delgadino, Gvalani and Pavliotis (Arch. Ration. Mech. Anal., 2021) to the hypoelliptic/hypocoercive case. We show that, in the absence of phase transitions, the two limits commute, and we calculate the covariance matrix of the limiting Brownian motion using the Green-Kubo/Kipnis-Varadhan formula. However, at low temperatures, and in the presence of phase transitions, the two limits may not commute. We demonstrate our findings by providing a detailed analysis of the diffusive-mean field limit for the $O(2)$ model in a magnetic field. Our analysis is based on the systematic use of recently developed hypocoercivity techniques, together with an appropriate linearization of the mean field McKean-Vlasov-Fokker-Planck partial differential equation.

We consider the Cauchy problem for the incompressible Euler-Vlasov-Fokker-Planck (Euler-VFP) system in the whole space \(\mathbb R^3\) near the global Maxwellian equilibrium. The Fokker-Planck operator and the particle-fluid drag dissipate the relative momentum but do not separately control the common particle-fluid momentum; in Fourier variables, this degeneracy occurs in the transverse momentum components. To recover the missing coercivity, we augment the classical four-moment compensator with a finite-rank skew-adjoint correction constructed from second-order Hermite modes. Combined with the cancellation between the kinetic and fluid drag terms and the incompressibility constraint, the resulting compensated Fourier energy yields a unique global classical solution for sufficiently small initial data $(u_0,f_0)\in H^N\times L_v^2(H^N)$, with $N\geq 4$. The high-order energy argument involves only spatial derivatives of the kinetic perturbation and requires no mixed \(x\)-\(v\) derivative estimates. We further construct a positive-order Lyapunov functional and establish the decay rate \((1+t)^{-1/2}\) for all positive-order spatial derivatives in the \(L^2\)-norm and for the corresponding pointwise-in-space norms, without any additional \(L^1\) integrability or low-frequency assumption on the initial data. Although no uniform algebraic decay rate is asserted for the zero-order energy of \((u,f)\), the directly dissipative variables \(u-J(f)\) and \(\{\mathbf I-\mathbf P_0\}f\) decay in the \(L^2\)-norm at the same rate, where $ J(f)=\int_{\mathbb R^3}v\sqrt M f\,{\rm d}v$ denotes the particle momentum and \(\mathbf P_0\) is the orthogonal projection onto \(\operatorname{span}\{\sqrt M,v_1\sqrt M,v_2\sqrt M,v_3\sqrt M\}\). To the best of our knowledge, these positive-order and zero-order decay estimates have not previously been established for the incompressible Euler-VFP system.

We derive large-mode asymptotic expansions for the eigenvalues and eigenfunction norms of the Chen-Libby perturbation problem about Falkner-Skan boundary-layer profiles. The analysis extends Brown's matched-asymptotic construction for the Blasius problem to Falkner-Skan solutions with positive wall shear – both favorable gradients and the upper branch of the mild adverse range. Although the outer and middle layers retain the same structure as in the Blasius case, the wall layer changes qualitatively when the pressure-gradient parameter $\beta$ is nonzero: the Falkner-Skan wall expansion singularly perturbs the inner Bessel problem at relative order $\Lambda^{-1/3}$, where $\Lambda$ is the large eigenvalue parameter. This produces a new wall-induced contribution $s^{-1/3}$ (where $s=n-1$ and $n$ is the large eigenvalue index) to the large-mode eigenvalue expansion, ahead of Brown's $s^{-1/2}$ correction. The new term vanishes in the Blasius limit, where Brown's ordering is recovered. The matched eigenfunction also yields asymptotic estimates for the Chen-Libby norms. The eigenvalue and norm formulae are compared with direct numerical shooting calculations for $\beta=1/2$, showing the expected asymptotic convergence and confirming the role of the new wall correction.

This article is devoted to the inverse source problem of uniquely determining a measure-valued source from sparse boundary measurements. The measurements considered consist of flux observations over a time interval at two distinct points on the boundary of the domain. The main objective of this work is to extend the existing literature on inverse source problems from sparse boundary measurements, which has so far been limited to point sources or L2 sources, to the identification of a general class of Radon measures. Our approach combines several analytical tools, including regularity properties, boundary representations, and the time analyticity of solutions to the diffusion equation with singular sources. Our theoretical analysis is complemented by a numerical study of the problem. In particular, we investigate the reconstruction of point sources and of a source supported on a curve, and present numerical experiments illustrating the recovery of such sources from sparse boundary flux measurements.

We prove that, for a given $C^{r,\beta}$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space $\mathbb{R}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,\alpha}$ for any $\alpha<\alpha_0=\min\{\frac{r+\beta}{2}, \frac{1}{1+d(d+1)/k}\}$. Our theorem recovers several previously known results as special cases. The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1, \frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available. Our threshold flexibility exponent $\alpha_0$ agrees with that previously obtained for the closely related Monge-Ampère system. As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4\alpha_0}{\alpha_0+1}$.

We establish sharp interior regularity for solutions to vectorial, nonlinear local-nonlocal systems.

We consider the Burgers equation with a scalar control acting through a fixed spatial profile, in a regime where the linearized system is not controllable. We identify natural conditions on the control profile leading to quadratic obstructions to small-time local null-controllability, quantified by negative Sobolev norms of the control of arbitrary integer order. We show that these conditions are related to iterated Lie brackets. The proof relies on repeated integrations by parts in the quadratic expansion combined with nonlinear remainder estimates. Finally, we construct smooth spatial profiles that produce obstructions at every negative integer order.

We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(\varepsilon\). Assuming a quantitative corrected \(H^1\)-estimate, a two-scale state class yields \[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+\Phi_{0,m_0}^2+\Phi_{1,m_1}^2\bigr) \] in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(\varepsilon)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K^{-1})\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((\varepsilon\sqrt N)^{-1}\) and \((\varepsilon^2\sqrt N)^{-1}\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(\varepsilon\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H^1\) errors as the microscopic scale is refined.

We show that the velocity solution $u$ to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields $\omega$ and $\thetaup$, $u=\omega+\thetaup$. Both $\omega$ and $\thetaup$ solve screened Poisson vector equations with right hand side respectively the rotational component in the Helmholtz decomposition of the external force, and the gradient of a harmonic scalar function $q$ called “solid pressure". Similarly, the fluid pressure $p$ is the superposition of the potential $\pi$ in the Helmholtz decomposition of the external force, and of $q$, $p=\pi+q$. It turns out that regardless the pressure $q$, $\omega+\thetaup$ is divergence-free and satisfies the generalized Stokes equations, except the normal component of the Dirichlet boundary condition. The role of $q$, which solves a boundary linear equation with a self-adjoint definite positive operator, is to constrain $\omega+ \thetaup$ to satisfy even the normal component of the Dirichlet boundary condition. The method is attractive for solving numerically the time dependent Navier-Stokes equations in two and three dimensions. It decouples the velocity from the pressure and solves the incompressibility constraint as a byproduct of unconstrained screened Poisson vector equations for $\omega$ and $\thetaup$, equipped with appropriate boundary conditions. In the two last sections we present two methods for solving numerically the generalized Stokes equations, and some results to demonstrate the efficiency of our approach.

We study the $L^p$ boundedness of the Riesz transform and the reverse Riesz inequality for degenerate elliptic operators of Grushin type. We prove full-range $L^p$ boundedness of the Riesz transform when the degenerate variable has dimension at least two, and obtain the sharp range in the one-dimensional weakly degenerate case, including the endpoint obstruction. In the strongly degenerate one-dimensional regime, we recover full-range boundedness, revealing a striking transition in the behavior of the singular set. The proof develops a reverse Hölder theory for Grushin harmonic functions near the singular set. The main ingredients are explicit Poisson and Green kernel constructions adapted to the Friedrichs extension and a harmonic annihilation method which isolates the critical part of the Riesz kernel. These techniques illuminate the mechanism behind both the boundedness and unboundedness phenomena, and yield essentially sharp reverse Riesz inequalities for the same class of operators.

In this paper, we study the Schrödinger–Bopp–Podolsky system \begin{equation*} \begin{cases} -\Delta u + V(x)u + \phi u = f(x,u), & \text{in } \mathbb{R}^3, -\Delta \phi + a^2 \Delta^2 \phi = 4\pi u^2, & \text{in } \mathbb{R}^3. \end{cases} \end{equation*} We consider the case where the potential \(V\) is indefinite so that the Schrödinger operator \(-\Delta + V\) has a finite-dimensional negative space. Under suitable assumptions on the potential \(V\) and nonlinearity $f(x,u)$, we prove the existence of nontrivial solutions via a local linking argument and Morse theory. Moreover, these solutions are shown to decay exponentially at infinity. Additionally, a ground state solution is obtained by minimization techniques. Finally, if \(f(x,u)\) is odd with respect to \(u\), we obtain an unbounded sequence of solutions using the symmetric mountain pass theorem.

In this paper, we study a system of nonlocal conservation laws motivated by traffic flow: a nonlocal version of the generalized Aw-Rascle-Zhang (GARZ) model. The nonlocality arises from downstream spatial averaging of the velocity by a one-sided kernel. We prove the existence and uniqueness of weak solutions for initial data of bounded variation via a fixed-point argument in the nonlocal velocity. We also establish stability with respect to the initial datum and an approximation of weak solutions by strong solutions in $L^1$. Under additional, physically meaningful assumptions on the velocity and the initial datum, we obtain either a maximum principle for the density or invariant-region estimates. Finally, we study the singular limit as the nonlocal kernel converges to a Dirac distribution. Indeed, under additional assumptions, convergence to the unique local entropy solution can be proved. Some numerical simulations are provided, and the paper concludes with a discussion of open problems.

Computer-assisted proofs of self-similar singularity formation for fluid equations often rely on numerically constructed approximate profiles. One effective approach to establishing stability of perturbations around a numerically constructed profile is to perform weighted energy estimates with singular weights near the singularity. However, the weighted norms require exact local vanishing conditions that are not automatically preserved by the equations nor the numerical construction. In this paper, we review an analytic low-rank correction method first developed in [ChenHou2023a,ChenHou2023b] to overcome this difficulty. The numerical step determines coefficients, rigorous bounds, and low-order defect modes in explicit global basis representations, while the required vanishing conditions are enforced analytically through low-rank corrections derived from Taylor expansions of the relevant quantities represented in a smooth basis. For completeness, we briefly review the singularly weighted estimates and a quantitative finite-rank perturbation method in the 2D Boussinesq / 3D Euler stability argument, where singular weights and the required vanishing order arise. Against this background, we formulate the local correction principle in a simplified setting, explain the correction of the residual error in numerical constructions of approximate space-time solutions and the stream function, and discuss its broader applicability to computer-assisted stability analysis for nonlocal PDEs.

This paper introduces a dynamic portfolio optimization framework for large institutional investors using Scientific Physics-Informed Reinforcement Learning (SciPhyRL). Formulated in continuous time over an extended state space that includes explicit cumulative costs, the approach leverages offline historical data to learn optimal, distribution-aware strategies. A core innovation reduces the optimization challenge to solving an HJB equation by projecting it onto observed trajectories as a pathwise Hamilton-Jacobi equation. This is solved directly from data using PINN in a single offline sweep, eliminating the need for traditional value or policy iteration. To make the method effective at practical short horizons, the control variable is recast from a continuous trading rate to a discrete target holding. This ensures signal-implied positions are reached immediately, while execution costs are evaluated against a microstructure-grounded quadratic price impact model. Evaluated on a $14$-asset ETF universe using an engineered oracle signal, the learned Gibbs policy yields substantial out-of-sample Sharpe ratio improvements over static and myopic baselines. The results demonstrate that the proposed framework successfully translates known signal quality into a robust, multi-period, and cost-aware allocation mechanism with strictly controlled volatility and turnover.

We study the optimal constants in higher-order Hardy inequalities on the lattice $\mathbb{Z}^d$. For each fixed $\ell \in \mathbb{N}$, we prove that the optimal constant $\mathcal{C}_\text{opt}^\ell(d)$ in $$ \sum_{n \in \mathbb{Z}^d} |\Delta^{\ell/2}u(n)|^2 \geq \mathcal{C}_\text{opt}^\ell(d)\sum_{n \in \mathbb{Z}^d} \frac{|u(n)|^2}{|n|^{2\ell}}. $$ satisfies $$ \lim_{d\rightarrow\infty}\frac{\mathcal{C}_\text{opt}^\ell(d)}{d^\ell} =2^\ell. $$ The proof is based on a Fourier reduction to a family of singular Hardy inequalities on the flat torus, involving the weight \[ \omega(x)^{-2\ell}, \qquad \omega(x)^2=\sum_{j=1}^d\sin^2\left(\frac{x_j}{2}\right), \] and zero average condition on admissible functions. We establish these torus inequalities by combining a ground state representation formula with a weighted integrated Bochner identity in an iterative scheme. The method yields explicit constants, defined recursively in the order $\ell$, and requires only the classical unweighted Poincaré inequality on the torus. The appearance of the limiting constant $2^\ell$ is particularly striking, as it suggests that, in the high dimensional regime, the optimizers are localized near the unit sphere $\{n\in\mathbb{Z}^d:|n|=1\}$ in $\mathbb{Z}^d$.

We investigate the ultradifferentiable regularity of solutions and vectors of a class of hypoelliptic differential operators first introduced by Hörmander. Our results extend results of Hashimoto–Morimoto–Matsuzawa and Bolley–Camus–Metivier.

We prove a Weyl law for the number of quasinormal modes (QNM) of a Schwarzschild black hole contained in a sector below the real axis. This requires introducing a new pseudodifferential operator calculus tailored to the study of semiclassical spectral problems near threshold energies. Elliptic theory in this calculus can be combined with the method of complex scaling to give uniform resolvent estimates near zero energy for operators that behave at infinity like a semiclassical Schrödinger operator with a repulsive inverse-square potential. Applied to the Regge-Wheeler potential, our methods imply the absence of high angular momentum QNM from a disc whose radius grows linearly with the angular momentum. Together with the asymptotic description of Schwarzschild QNM recently obtained by Hitrik and Zworski, this shows that the number of QNM contained in a small sector below the real axis and with modulus bounded by $\lambda$ grows as $C\lambda^3$. We also study the effect of cutting off the Schwarzschild resolvent away from the event horizon and show that such a cutoff does not lead to any pole cancellations.

Establishing the global-in-time smoothness of solutions to the 3D Navier-Stokes equations (NSEs) with large initial data remains a long-standing open problem. In this paper, we prove that the 3D NSEs driven by a suitably chosen transport noise – a physically motivated stochastic perturbation – admit global-in-time smooth solutions with high probability. This regularizing effect holds uniformly for initial data in arbitrarily large balls of subcritical function spaces with positive smoothness.

We investigate a second-order Modica-Mortola functional of the form \[ E_\varepsilon[u] := \int_\Omega \frac{1}{\varepsilon} W(x, \nabla u) + \varepsilon|\nabla^2 u|^2 \, dx, \] which models solid–solid phase transitions in heterogeneous media. We neglect the assumption of frame indifference in the elastic energy density $W$ while allowing for space-dependent, pointwise-compatible wells. Under suitable assumptions on $W$ and regularity conditions on the associated rank-one connection, we prove that $E_\varepsilon$ $\Gamma$-converges to a local interfacial energy defined for suitable laminate-type configurations.

We study the $L^p$ Neumann problem for the Stokes system on bounded Lipschitz domains in $\mathbb R^n$ with $n\ge 2$. Our main contribution is to introduce a nonlinear gradient quantity – namely, the linear gradient weighted by a suitable power of the pair $(\nabla u,\phi)$ – in place of the standard linear gradient used in previous work by Geng and Shen (2025). This new approach allows us to establish a global second-order estimate, which in turn yields an improved reverse Hölder inequality and extends the known range of solvability for convex domains, particularly improving the upper bound for $n\ge 3$. Beyond the convex setting, our method also applies to semi-convex domains. Moreover, for more general Lipschitz domains, we prove solvability under a smallness condition on the second fundamental form of the boundary, assuming the boundary has second-order derivatives in the weak-type Lorentz spaces $W^2L^{n-1,\infty}$ for $n\ge 3$, or $W^2L^{1,\infty}\log L$ for $n=2$. In particular, our results cover all $W^{2,q}$ domains with $q>n-1$.

We study local integrability of gradients of singular interaction kernels in aggregation equations. Suppose near the singularity that $|\nabla K|\asymp|\nabla f|\,|f|^{-(\kappa+1)}$, where $\kappa>0$ and $f$ is real-analytic with $f(0)=0$. A real log-resolution of $f$ and its Jacobian ideal gives the exact criterion $\nabla K\in L^p_{\mathrm{loc}}$ if and only if $p<p^*(\kappa)$, where \[ p^*(\kappa)=\min_i\frac{a_i+1}{(\kappa+1)\nu_i-M_i}. \] Here $\nu_i$, $M_i$, and $a_i$ are the vanishing orders of $f$, those of its Jacobian ideal, and the discrepancy exponents. Moreover, $\operatorname{rlct}_0(f)/(\kappa+1)\leq p^*(\kappa)$, with equality exactly when an RLCT-computing divisor has $M_i=0$; in particular, the inequality is strict for isolated zeros in dimension at least two. For quasi-homogeneous singularities with isolated real zero, $p^*(\kappa)=|w|/(\kappa d+\max_i w_i)$. Finally, if $p^*(\kappa)>1$, a suitable global truncation gives $\nabla K\in L^1(\mathbb R^n)$, yielding the kernel hypothesis in the Morrey-space well-posedness theorem of \cite{suleiman2023existence}.