Papers

We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as compared to the growth due to the forcing term, solutions may propagate or vanish as time passes. We prove new results concerning this dichotomy that include the critical case where diffusion and reaction are of the same order. If the initial datum possesses some symmetry (invariance under a cohomogeneity one subgroup of the group of isometries of the hyperbolic space), the problem reduces to a unidimensional one. In the case of propagation, the solution to this unidimensional problem converges in shape to an Euclidean traveling wave of minimal speed in an appropriate moving frame. The choice of this frame depends on the subgroup of isometries (elliptic, hyperbolic or parabolic) under which the initial datum is invariant. In contrast with the Euclidean case, the asymptotic spreading speed (in due coordinates) depends on the dimension, while the coefficient of the logarithmic correction in the location of the front does not, no matter the underlying isometry.

We analyze spectra and the Riesz property of spectral projections of non-symmetric perturbations of self-adjoint operators with eigenvalues having arbitrary multiplicities, including infinite ones. In particular, we establish the Riesz property for perturbations of the multi-dimensional harmonic oscillator, Landau Hamiltonian and Laplace-Beltrami operator on a sphere by complex-valued $L^r$-potentials if $d/2 < r < \infty$.

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean field control. In addition to reviewing the main ideas and known results, we present two new contributions. First, we provide a variational construction of generalized characteristics via a minimizing movement scheme; by taking the weak limit of approximate solutions and using Young measure compactness, we show that the limiting curve satisfies the generalized characteristic differential inclusion. Second, we lift the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and study its invariant probability measures. We prove that the only invariant measures of the GHGF semi-flow that attain the critical value \(c[H]\) are precisely the projected Mather measures, thereby giving a new dynamical characterization of Mather's minimal measures as well as Mañé's critical value. Finally, we discuss a number of open problems that arise from the GHGF perspective, including questions on uniqueness of strict singular characteristics, rectifiability of cut loci, stability under perturbations, contact Hamiltonian systems, vanishing noise limits, and extensions to non-convex or low-regularity Hamiltonians. These problems highlight the deeper connections between singular dynamics, ergodic theory, optimal transport, and geometric analysis, and indicate directions for future research.

This paper investigates the global well-posedness and large-time behavior of 3D incompressible active liquid crystals under constant activity, modeled by a coupled system of forced incompressible Navier-Stokes equations for the velocity and a parabolic system for the $Q$-tensor order parameter. By employing refined commutator estimates, the existence and uniqueness of global strong solutions are proved for small initial data $(Q_0,u_0)\in H^{s+1}\times H^s$ $(s\geq 2)$ with activity $c>c_\star$, which improves a previous result in \cite{active-limit}. In addition, if the initial data further belong to $L^1$ and $s\geq 4$, we obtain a mixing decay estimate on $\|\partial^kQ(t)\|_{L^2}$ that combines both an extra exponential decay factor at a rate proportional to $(c-c_\star)\Gamma$ and the optimal algebraic decay rate that coincides with that of the heat kernel, where $k\leq s-1$. This result reveals that, in the high activity regime, active nematics become isotropic with an activity-dependent exponential convergence rate, and the estimate is stable in the infinite rotational viscosity limit, as $\Gamma\rightarrow 0$. Meanwhile, the sharp decay estimate on $\|\partial^ku(t)\|_{L^2}$ is also derived for $k\leq s-2$ with an additional initial assumption. The proof is established via a combination of the Green's function method and the time-weighted energy method. To the best of our knowledge, these results are the first reported for active/passive nematic liquid crystals within the Beris-Edwards framework, and the enhanced decay effect of the orientational field is essentially derived from the free energy. Furthermore, in the passive setting, our result implies the phase transition of thermotropic liquid crystals at high temperatures.

We study a finite-time source-driven transition in axisymmetric Euler flow with swirl. For an explicit smooth compact datum, we prove exact source, moment-return, fixed-label transport, and response identities. For the same frozen finite reference, interval arithmetic gives continuum derivative bounds and a terminal scalar-trace enclosure. Under the remaining global comparison hypotheses, these estimates imply contraction of two fixed material labels and a finite-radius outgoing level set. We state separately the unresolved steps: global defect enclosure, sufficient next-block admission, cylindrical whole-space realization, and repeated regeneration from one datum. No whole-space Euler singularity is claimed

This paper proves several topological results for smooth gradient Ricci shrinkers. We establish upper bounds for the Betti numbers, a vanishing theorem for cohomology, and a dichotomy for the number of ends. We also prove a full Hodge theorem for a large class of shrinkers. The methods are based on weighted $L^2$ cohomology and extend to self-shrinkers of the mean curvature flow.

We study the Cauchy problem for the three-dimensional isentropic compressible ideal (inviscid and non-resistive) magnetohydrodynamic equations with velocity damping on the periodic torus $\mathbb{T}^3$. The system admits a steady equilibrium consisting of a constant density $\bar{\rho}$ and a uniform background magnetic field $\omega\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\bar{\rho},\mathbf{0},\omega)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, and if $\omega$ satisfies a Diophantine condition, then the system admits a unique global smooth solution. Moreover, the perturbations decay algebraically in time. To the best of our knowledge, this is the first global well-posedness result for the multi-dimensional isentropic compressible ideal MHD system. The proof reveals a hidden dissipation mechanism: although neither the density equation nor the magnetic field equation contains explicit diffusion or damping, the coupling between the velocity and the magnetic field through the background field $\omega$, combined with a Diophantine–Poincaré inequality, generates effective dissipation for both the density perturbation and the magnetic field perturbation, which together with the velocity damping yields global regularity and time decay.

We investigate local minimizers of Ginzburg–Landau-type functionals in dimension $n\geq 3$ that satisfy logarithmic energy bounds, assuming the potential has a vacuum manifold with a finite fundamental group. We show that the normalized energy measures converge to an $(n-2)$-rectifiable measure associated with a stationary varifold, with quantized density determined by the homotopy classes of the vacuum manifold. Away from the support of the $(n-2)$-rectifiable measure, the minimizers converge strongly in $H^1_{\text{loc}}$ to a minimizing harmonic map, which is smooth outside an $(n-3)$-rectifiable singular set.

This paper is devoted to the study of existence, uniqueness, stability, and monotonicity of traveling wave solutions to the following parabolic-elliptic chemotaxis system with logistic type source \begin{equation}\label{E:main-abstract-eq}\tag{CM} \begin{cases} u_t=u_{xx}-\chi(u^m v_x)_x +u(1-u^\alpha),\quad &x\in\mathbb{R}\cr 0=v_{xx}-v+u^\gamma,\quad&x\in\mathbb{R}, \end{cases} \end{equation} where $m,\alpha,\gamma\ge 1$ and $\chi\in\mathbb{R}$. System (CM) can be used to describe the evolution of a biological species influenced by a chemical substance produced by the species itself. In this context, the function $u$ denotes the population density of the biological species, and $v$ denotes the concentration of the chemical agent. Traveling wave solutions of (CM) connecting the two constant solutions $(1,1)$ and $(0,0)$ are among important types of solutions, which characterize the front propagation phenomena in (CM). The existence of such traveling wave solutions to (CM) with $m=\alpha=\gamma=1$ has been studied in several papers. However, there is little study on the uniqueness, stability, and monotonicity of traveling wave solutions of (CM) in literature and there is also no study on the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for general $m,\alpha,\gamma\ge 1$. In the current paper, we prove the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for any $\chi\le 0$ with speed $c$ large than some number $c^*_{\chi,m,\gamma}$, or for $0<\chi<1/2$ with any speed $c>2$. We prove that the traveling wave solutions are monotone when $\chi\le 0$. We also prove the uniqueness and stability of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ when the speed $c$ is larger than some number $c^{**}_{\chi,m,\alpha, \gamma}(\ge c^*_{\chi,m,\gamma})$.

In this paper, we investigate the global existence of spherically symmetric strong solutions with large initial data to an initial-boundary value problem of the multidimensional isentropic compressible Navier-Stokes-Korteweg system in an unbounded exterior domain. We consider the case when the pressure $p(\rho)=\rho^\gamma$, the viscosity coefficients $\mu(\rho)$ and $ \lambda(\rho)$ satisfy either $\mu(\rho)=\tilde{\mu}, \lambda(\rho)=\tilde{\lambda}\rho^\alpha$ or $\mu(\rho)=\tilde{\mu}\rho^\alpha, \lambda(\rho)=\tilde{\lambda}\rho^\alpha$, and the capillarity coefficient $\kappa(\rho)=\tilde{\kappa}\rho^\beta$, where $\alpha,\beta,\gamma \in \mathbb{R}$ are parameters, and $\tilde{\mu},\tilde{\lambda},\tilde{\kappa}$ are given real constants. Under suitable restrictions on the parameters $\alpha,\beta$ and $\gamma$, we establish the global existence and uniqueness of spherically symmetric strong solutions. The proof relies on the radically weighted energy method combined with the technique developed by Y.~Kanel'\cite{28}.

We derive a unified vorticity–stream formulation $(Bm)$ for two parity-reduced inviscid systems in the meridian plane: the 2D inviscid Boussinesq equations $(m=1)$ and the 3D axisymmetric Euler equations with swirl $(m=2)$. In the Boussinesq case we set $\Theta=\vartheta/r$ and write $\Theta=u^2$ only when a smooth square-root branch has been fixed; equivalently, one may keep the scalar variable $\Theta$ throughout. In the squared radial variable $q=r^2$, the two cases are encoded by the same parameterized system with $m=1,2$. At the boundary $q=1$, a Taylor expansion gives an exact boundary jet: the transport equations close on the boundary, while the elliptic relation also contains the next normal jet $\varphi_{qq}(x,1,t)$. If the boundary jet is closed by the first-order Taylor truncation $\varphi_{qq}(x,1,t)=0$, it reduces to a closed unified $(1+1)$D system $(Q0)$ with the local boundary velocity law $u=-(m+2)^{-1}\omega$. We prove finite-time blow-up for this closed Hou–Luo type model on a periodic interval by a Riccati argument in the spirit of Choi–Hou–Kiselev–Luo–Šverák–Yao. The theorem is therefore a blow-up result for the closed boundary-jet model, not for the unrestricted Boussinesq or Euler systems.

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter. Consequences include an optimal two-term lower bound for the first Dirichlet eigenvalue in terms of area and perimeter. We also prove a Cheeger-type inequality with an explicit best constant considered by Parini. To prove these conjectures we propose a new method for proving Dirichlet eigenvalue inequalities on triangles. Our method is based on a new computable lower bound for second-order directional shape derivatives under vertex perturbations. It also uses validated finite-element error estimates and recently developed analytic estimates for eigenvalues of nearly degenerate triangles. The method is not specific to the functionals considered in this paper and it can be used to prove various other eigenvalue inequalities on triangles.

Existence and uniqueness of strong solutions to a barotropic compressible fluid–viscoelastic shell interaction system have recently been established on a finite time interval. A natural question is whether such solutions can be continued globally. In this work, we derive a continuation criterion for this coupled system. Our analysis is based on an energy estimate at the level of material acceleration, derived under Serrin-type and Beale–Kato–Majda-type control assumptions. While in the incompressible setting, such control is sufficient to prevent finite-time blow-up, in the compressible regime it does not by itself ensure propagation of the full regularity required for strong solutions. To obtain a genuine continuation criterion, we impose a Beale–Kato–Majda Lipschitz-type control on the density and velocity gradients with stronger time integrability. In combination with the control framework underlying the acceleration estimate, we close a higher-order energy estimate and thereby prevent loss of strong-solution regularity. Consequently, the solution can be extended beyond a potential blow-up time, provided that the corresponding control norms remain finite. We further establish a weak-strong uniqueness principle for the system under the above conditional regularity criterion.

We consider a fractional variant of Maxwell's equations, where the electric and magnetic fields are modeled as two-point fields. To formulate the system, we introduce a fractional curl operator that is compatible with the fractional divergence operator, ensuring the divergence-free condition. A key ingredient is a projection map $\Pi$ that reduces two-point fields to one-point fields. We also define a new fractional Sobolev space whose elements enjoy a fractional Helmholtz decomposition and observe that the projection $\Pi$ is a bijection in this space, which allows us to reformulate the problem entirely in terms of one-point fields. We then prove the well-posedness of the equations in one-point fields in weighted fractional Sobolev spaces, and deduce a corresponding well-posedness result for the two-points fractional Maxwell system. This constitutes a first necessary step towards the resolution of a scattering inverse problem for the fractional Maxwell's equations, which will be the topic of future work.

We study hyperbolic side-wall packets for the axisymmetric Euler equations with swirl in a periodic cylinder. In the exact odd class, $\Gamma=ru^\theta$ and $G=\omega^\theta/r$ satisfy $D_t\Gamma=0$ and $D_tG=r^{-4}\partial_z(\Gamma^2)$, and the symmetry fixes the boundary point at which the relevant swirl and vorticity gradients are measured. We derive the five-dimensional side-wall Green-kernel expansion, its leading signed hyperbolic kernel, packet–core coupling, off-diagonal shear estimates, anisotropy and source-curvature identities, and a maximal conic-score formulation. Under a common admissible packet bootstrap these estimates imply the Dini system $D^+\mathfrak M\ge cB^2$, $B'\ge c\mathfrak M B$, and hence finite-time blow-up of the comparison amplitudes. The result is a conditional amplification criterion: it does not prove that one smooth Euler trajectory preserves all packet-dominance, tail, shear, anisotropy, and validity hypotheses until the comparison blow-up time. Establishing or refuting that persistence is the remaining problem for this mechanism.

We study a general linear parabolic problem for Petrovskii parabolic differential system in Sobolev anisotropic distribution spaces of generalized smoothness. Slowly varying functions are used to characterize supplementary generalized smoothness that cannot be determined by number indexes. We prove that this problem induces topological isomorphisms on appropriate pairs of such spaces. As an application, we give sufficient and necessary conditions for the problem solutions to have prescribed generalized regularity expressed in terms of these spaces. Their use allows obtaining exact conditions for indicated generalized partial derivatives of the solutions to be continuous.

We prove a generalization of the Maz'ya-Shaposhnikova formula in the case $p=2$ for functions that may not belong to ${L^2}(\mathbb{R}^d)$ and, thus, might not vanish at infinity. By introducing a notion of mass at infinity, we explicitly characterize the limit as $s\to0^+$ of Gagliardo seminorms localized on a bounded Lipschitz domain $\Omega$. By `localized', we mean here that we account only for interactions involving at least one point in $\Omega$. The identified limiting functional provides a unifying framework to link the classical Maz'ya-Shaposhnikova formula and the asymptotics of nonlocal perimeters. On the one hand, it reduces to the classical $L^2$ norm for functions that are globally integrable on $\mathbb{R}^d$. On the other hand, it recovers the pointwise limit of $s$-fractional perimeters when evaluated on characteristic functions of sets. We further show that the same functional encodes the asymptotic behavior of Gagliardo seminorms in the sense of Gamma-convergence with respect to the weak-$L^2$ topology. Finally, we provide an extension to the setting of metric measure spaces.

We study the stability of traveling wave solutions to the Burgers–Hilbert equation on $\mathbb{T}$ in the regime of small frequency $\omega$ and large wave speed $c$. For $\omega = 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers–Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states.

We establish maximal local $L^p$-well-posedness, for $p>3$, for the stochastic tamed Navier–Stokes equations on $\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener noise and a compensated Poisson random measure. For divergence-free initial data $ u_0\in L^p\bigl(\Omega,\mathcal F_0; L^p(\mathbb R^3;\mathbb R^3)\bigr), $ we prove existence and pathwise uniqueness of a local strong solution with $L^p$-valued càdlàg trajectories and the local $L^p$-energy regularity. For solutions driven by the same noises, we establish localized Lipschitz dependence on the initial datum in the path supremum and space–time norms. The discontinuous forcing makes the whole-space Gaussian construction non formal as stopping levels may be overshot by jumps, while convergence of the compensated-Poisson term requires simultaneous control of its quadratic and $p$th integrability modes. We resolve these difficulties by a jump-compatible localization based on strict pre-exit bounds and predictable left limits. We further establish bounded-time restart and stochastic pasting on the prescribed stochastic basis. The resulting family of attainable lifetimes is upward directed and yields a unique maximal local strong solution, which inherits the localized dependence estimate. Continuation criteria, blow-up alternatives, and global well-posedness under stronger finite-energy hypotheses are treated in the companion Part~II.

In this paper, we prove a sharp quantitative stability result for the affine fractional \(L^2\)-Sobolev inequality in \(\dot H^s(\mathbb R^n)\), \(0<s<1\), introduced by Haddad–Ludwig (Math. Ann. 388 (2024), 1091–1115). In particular, we identify the kernel of the affine Hessian, determine the sharp local spectral gap, and show that the optimal global stability constant is strictly smaller than the corresponding local spectral value.

We study a free boundary problem for the three-dimensional steady compressible Euler equations in an infinitely long circular cylinder. The free boundary is a contact discontinuity separating an axisymmetric rotational subsonic flow and an axisymmetric potential subsonic flow, neither of which is prescribed a priori. The pressure continuity condition couples two unknown Euler phases through an unknown interface, leading to a genuinely two-phase free boundary problem. Using a Helmholtz decomposition, we reformulate the pressure continuity condition as nonlinear boundary conditions for the Helmholtz variables and develop a coupled iteration framework that simultaneously determines the free boundary and the two flow fields. Uniform estimates independent of the truncation length allow us to pass to the infinite-length limit and construct global solutions. We further establish the downstream asymptotic behavior of solutions, showing that the contact discontinuity becomes asymptotically cylindrical and that the radial velocity vanishes at infinity. To the best of our knowledge, this provides the first existence result and asymptotic characterization for a genuinely two-phase free boundary problem involving a contact discontinuity between an unknown rotational subsonic flow and an unknown potential subsonic flow in a three-dimensional infinitely long cylinder.

We study the singular semilinear equation $-Pu = \frac{f}{u^\gamma}$ on a bounded domain $\Omega$ with Dirichlet condition $u \equiv 0$ on $\partial \Omega$ , where $P$ is a second-order elliptic differential operator in nondivergence form. We obtain the existence of a solution under the assumptions that $\Omega \in C^{1,1}$ and $P$ has $C^1$ coefficients, as well as the uniqueness of solutions in $L^1(\Omega)$, under the assumptions that $\Omega \in C^2$ and $P$ has $C^2$ coefficients. Our proofs are based on a novel combination of tools, such as recently obtained nonlinear variants of Gagliardo–Nirenberg inequalities, estimates of Green functions, and new variants of Kato-type inequalities.

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\Delta+(-\Delta)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad (x,t)\in \mathbb{R}^{d}\times(0,\infty), \] where $f:\mathbb{R}\to\mathbb{R}$ exhibits exponential growth at infinity and satisfies $f(0)=0$. We establish local well-posedness in a suitable Orlicz space in the case where $f(u)\sim e^{|u|^p}$ as $|u|\to\infty$, with $p>1$. We further prove the existence of global solutions for small initial data under the assumption that $f$ satisfies the growth condition $|f(u)|\sim |u|^m$ near the origin. Moreover, we derive large-time decay estimates in Lebesgue spaces, showing that the behavior of the nonlinearity near the origin determines the decay rate of solutions and highlights a unique asymptotic transition that bridges local and non-local diffusion theories.

Stein Variational Gradient Descent (SVGD) is a widely used in practice algorithm for scalable sampling with deterministic particle updates. We study its behavior in the singular limit where the kernel bandwidth tends to zero. In this regime, we show that the nonlocal SVGD dynamics converge to a local evolution equation that can be formally interpreted as a Wasserstein gradient flow with quadratic mobility. We analyze this singular limit in two settings: integrable kernels and weighted kernels. In the weighted case, the proof is supported by recently established Stein-log-Sobolev inequalities, which provide the necessary functional control. Overall, our results clarify how SVGD collapses from a nonlocal interacting particle system to a local gradient-flow dynamics as the kernel concentrates.

In this paper we study a mass-constrained free boundary problem modeling cell polarization, in the regime where the mass is small. In the generic case of a signal with nondegenerate maxima, we prove that the solution converges locally to a global, integrable solution to an obstacle problem in the plane. We further show that the interface of the solution to the limit problem is an ellipse, the equation of which is explicit. We also study some cases where the signal has degenerate maxima, highlighting a variety of possible behaviors.

Using an approach similar to arXiv:2409.15460, we give a new proof of the nonlinear stability of de Sitter space as a solution to the Einstein vacuum equations with positive cosmological constant in $n+1$ dimensions, with $n\geq3$. Using the gauge freedom of the equations, we are able to prove a precise expansion of the perturbed spacetime at the conformal boundary. In $n=$ odd spatial dimensions, the conformally rescaled metric is smooth up to the future conformal boundary and in $n=$ even spatial dimensions it is smooth if and only if the obstruction tensor of the boundary metric vanishes; if not, then the conformally rescaled metric is log smooth at the boundary. These results also hold for asymptotically de Sitter spaces. Using the results of Fefferman and Graham (1985, Conformal invariants), arXiv:0710.0919, arXiv:1705.09674 and arXiv:2311.02739, the structure of our expansion allows us to establish a 1-1 correspondence between solutions to the Einstein vacuum equations close to de Sitter space and scattering data prescribed on the conformal boundary in general dimension.

We treat the convergence of Carleman linearization of nonlinear evolutionary equations through the approximation theory of strongly continuous semigroups, by Carleman embedding the underlying nonlinear semigroups as linear semigroups. Linear semigroup theory then lets one replace the norm constraint on the convergence of Carleman linearization in the form used by quantum algorithms for a class of semi-discretized evolution equations by a dissipativity constraint, simplifying arguments for convergence. Applying Trotter-Kato approximation theorem to the linearized semigroup realizes the semigroup as a limit finite dimensional operator exponentials, reducing the question of convergence rate of Carleman linearization to that of the Trotter-Kato approximation. We then examine convergence of the Carleman linearization as the operators become unbounded, treating the hyperviscuous Burger's equation as an example. Next we consider the perturbation theory of the Carleman semigroup and obtain conditions when polynomial nonlinearities correspond to the Carleman linearized semigroup being a $1$-integrated semigroup, so convergence is implied by variants of Trotter-Kato approximation for integrated semigroups.

Given a complete doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we approximate harmonic functions on a bounded domain $\Omega$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(\Omega)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(\Omega)$ is obtained as a $\Gamma$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs.

We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $\Omega\subset\mathbb R^d$, with boundary data given by the indicator of a closed set. At a smooth boundary point where the data vanish, we study the upper and lower inward normal Dini derivatives as $p\downarrow1$. We give sufficient geometric conditions for these derivatives to be of order $(p-1)^{-1}$, and other conditions under which they decay exponentially in $1/(p-1)$. Whether explosion or decay occurs is not determined locally: on one fixed planar domain with real-analytic boundary, changing the data on a set of arbitrarily small total length, uniformly separated from the observation point, changes explosion into exponential decay. We also exhibit a critical example of a cylinder in $\mathbb R^{d+1}$ where the inward normal derivative is of order $\sqrt{d/(p-1)}$.