Papers

We investigate an inverse coefficient problem for an augmented Keller–Segel model arising in the description of phototactic and chemotactic population dynamics. The spatiotemporal evolution of the population density is governed by an advection-diffusion equation in which both the diffusion coefficient and the advection coefficient depend on space and time. Due to the underlying intracellular adaptation mechanism, even in a time-independent external environment, the diffusion coefficient \(D(x,t)\) and the scalar advection coefficient \(K(x,t)\) remain time-dependent and relax exponentially toward their respective steady-state profiles. Importantly, their relaxation is governed by the same rate, determined by the intracellular adaptation dynamics. From internal measurements of the density, we establish conditional Lipschitz stability for recovering either coefficient when the other is known, and derive cross-sensitivity estimates quantifying the compensation between diffusion and advection perturbations that produce the same density data. We further identify a finite-time separation between the transient and steady regimes: the full dynamics admit an exponentially accurate frozen-coefficient approximation at late times. When the steady-state coefficients are known, we also prove unique identification of the relaxation rate from the long-time behavior of the solution. Finally, we derive an explicit linearization of the forward map and, motivated by the temporal separation above, develop a two-stage gradient-based reconstruction strategy. Numerical experiments demonstrate the feasibility of the proposed method.

We study the instability of the naked singularities arising in the spherically symmetric self-similar collapsing of the Einstein–Euler system under gravitational perturbations. We show that small $C^{1,\alpha}$ perturbations (without any symmetries) of the initial conformal metric lead to trapped surface formation. One key point is that the speed of sound is strictly slower the the speed of light, so the gravitational radiation can become sufficiently concentrated to form a trapped surface before the fluid develops any singularity.

For each positive integer $N$, we construct global spectral group actions for the defocusing and focusing nonlinear Schrödinger hierarchies on initial-data classes containing $W^{2N+1,\infty}(\mathbb R)$. For $N\ge2$, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato–Segal–Wilson and Kotani frameworks to matrix Dirac systems; a homogeneous tau-function identity supplies the focusing positivity. No decay, smallness, periodicity, arithmetic condition, or prescribed spectral background is required for the initial-data inclusion. As a principal application, we prove global well-posedness on $M_{\infty,1}^{5}(\mathbb R)$ for both signs, with finite-time bounds and locally Lipschitz dependence on initial-data norm balls. The passage to this Banach-space flow uses near-real-axis Weyl asymptotics and continuity of normalized Toeplitz operators on compact initial-data sets to obtain uniform spatial bounds from five bounded derivatives. In the defocusing case, the flow extends to $M_{\infty,1}^{5}(\mathbb R)+H^1(\mathbb R)$. Further consequences include quantitative approximation, preservation of spatial Bohr almost periodicity and its frequency module, conjugacy of spatial translation hulls, and preservation of the full complex Dirac spectrum and spatial transfer growth rates.

In this paper, we study a class of degenerate mean field games (MFG, for short) systems with Hormander diffusion, in which the typical agent can move only along admissible direction. We establish the well-posedness of the MFG systems in intrinsic Holder spaces, which describes the Nash equilibria for a differential game with infinitely many small players. The analysis builds upon degenerate parabolic equations defined on the tours induced by Hormander vector fields without any group structure, for which we develop a global regularity theory for the first time in this general setting. A central difficulty arises from the anisotropic geometry of the state space, which induces non-commutativity and inhomogeneity of the underlying vector fields. Instead of the classical fundamental solution framework, we present an alternative method to derive a priori Schauder estimates for general Hormander degenerate parabolic equations via Campanato spaces.

We prove a boundary analogue of the rigidity theorem for Ricci-pinched three-manifolds under a superquadratic volume-growth assumption. More precisely, we show that a complete connected three-manifold with convex boundary, pinched Ricci tensor, and pinched second fundamental form is isometric to the Euclidean half-space. The proof is potential-theoretic. We construct and study mixed Dirichlet–Neumann $p$-capacitary potentials, establish monotonicity formulas for their free-boundary level sets, and use a weak Gauss–Bonnet formula adapted to this setting.

We prove that every three-dimensional parallelohedron $P$ satisfies \[ \mathcal H^2(\partial P)\ge 12\sqrt2\,r(P)^2, \] where $r(P)$ is the inradius of $P$, with equality if and only if $P$ is similar to the regular rhombic dodecahedron.

We establish a concentration–compactness theory for a one-dimensional singular Moser-type functional with logarithmic weight function $$\sup\limits_{v\in E }\int_0^1 e^{\left(\log\frac{e}{s}+\gamma\log\log\frac{e}{s}\right)\frac{v^2(s)}{s}}ds<\infty,~\gamma\in[0,1],$$ where $$E:= \left\{u \in H^{1}(0, 1): u(0) = 0, \int_0^1 |u'|^2dx=1\right\}.$$ In particular, we prove compactness for all normalized sequences in the subcritical range $\gamma<1$, thereby resolving an open problem proposed in [J. M. do Ó, B. Ruf and P. Ubilla, CVPDE 2023]. We also study the critical Bliss-Moser inequality: \begin{equation}\nonumber \sup\limits_{v\in E }I^c(v)< +\infty,\quad I^c(v)=\int_0^1 e^{\left(\log\frac{e}{s}+\log\log\frac{e}{s}+cs\right)\frac{v^2(s)}{s}}ds. \end{equation} For any $c\in(-\infty,+\infty)$, we determine the sharp concentration level $$\sup_{v_j\in E,v_j~\text{is~NCS}}\limsup\limits_{j\to\infty} I^c(v_j)=1+e^2,$$ and obtain explicit sufficient conditions for attainability under $c>c^*$ and non-attainability under $c<-c_*$. This results describe how linear perturbations govern the transition between compactness, concentration, and attainability at the critical level.

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming–Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

We study the large-time behavior of the displacement for wave equations with non-integrable speeds and zero initial displacement. For initial velocities in $L^2\cap L^1$, we derive an upper bound estimate in the $L^2$ norm, with the rate determined by the propagation speed and its time integral. Under stronger assumptions on the propagation speed and the non-trivial mean condition on the initial velocity, we derive a lower bound estimate of the same order without using an explicit formula for the Fourier multiplier. The estimates apply to polynomial, logarithmically modified, log-periodically oscillating, and exponential-type speeds.

This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $\omega$ for which there exists a nontrivial incident $\omega$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).

In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type \[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{in } B_1 \subset \mathbb{R}^{n}, \] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \[ \mathscr{H}(x,\xi):= \langle \mathcal{B}(x),\xi\rangle+\varrho(x)|\xi|^{\sigma}, \quad 0< \sigma\leq2, \,\,\, \sigma \neq 1, \] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{for} \quad 0<m<1. \] Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($\sigma \neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.

We study local bifurcations from the single-mode traveling wave family of the filamentation equation on the torus, within the real positive-frequency Sobolev space $X^s$. A Fourier null structure removes the apparent derivative loss and makes the traveling wave map real analytic for $s>3/2$. Its linearization splits into finitely many coupled two-mode blocks, a carrier block, and a diagonal high-frequency tail. For negative temporal frequency, we determine the critical values generated by both the finite coupled blocks and the tail. At every non resonant finite-block critical value, Lyapunov-Schmidt reduction yields a locally unique real-analytic branch, with vanishing linear correction to the bifurcation parameter. Moreover, each finite tail critical value produces a simple local branch. These values accumulate at a parameter where the linearization ceases to be Fredholm. The accumulation point is nevertheless a bifurcation point in the usual topological sense. For $\sigma=1$, a symmetry in the first Fourier mode produces an exact two-mode vertical branch, including the exceptional non-Fredholm case $k=2$.

This paper investigates the wave propagation for a non-conservative compressible two-fluid model with unequal pressures in R3. This model is novel and fundamentally distinct from the classical single-phase compressible Navier-Stokes equations proposed by Liu-Wang [28]. This work fills a significant gap, as prior results largely apply only to conservative or partially conservative systems with specific Green's function cancellations. The non-conservative nature and the coexistence of two distinct-speed Huygens waves demand dedicated nonlinear coupling analysis and add further complexity. Our study reveals fundamentally different physical behaviors of this non-conservative two-phase model under equal and unequal interfacial pressures. For equal pressure, the model admits a stationary diffusion wave comprising a single Huygens wave, analogous to classical Navier-Stokes equations. In sharp contrast, under unequal pressure, the stationary diffusion wave consists of two Huygens waves propagating at different speeds–a novel phenomenon absent in the classical Navier-Stokes context.

We investigate the free boundary propagation of a non-linear degenerate parabolic problem with both fourth and second order non-linearities, which may be used to model viscous thin-film flow in the regime of weak slippage with an additional contributions from a singular disjoining pressure brought forth by inter-molecular forces. By carefully treating the individual scaling behaviors of the non-linearities, we obtain asymptotically optimal upper and lower bounds on the rate of forward support propagation. These rates show that the fourth-order operator dominates the propagation rate on short time scales, but the long-term rate of propagation is controlled by the second-order operator. We estimate further the time at which this trade-off in propagation rate occurs. Lastly, we examine sufficient conditions to ensure that the free boundary does not immediately advance in a given direction, leading to a detailed picture of how the competition between the second/fourth-order operators influence the free boundary dynamics of solutions.

We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the $H^1$-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order $s \geq 2$ and derive sublinear convergence rates in the $H^1$-norm of the form $O(k^{-\zeta})$, where $\zeta\in(1/2,1]$ depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order $s = 1$ with the Shannon and Spence entropies. This yields linear convergence of the form $O(\varrho^k)$ for some $\varrho\in(0,1)$.

We establish interior Hessian estimates for $k$ convex solutions of $\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}=1$, where $1 \leq l < k < n$, $k-l \in \{1,2\}$, and $2k > n$.The proof combines a concavity inequality with a pointwise doubling argument.

This paper investigates the stability of strong global and exponential attractors for a semilinear beam equation with memory and fractional damping, where $\alpha\in[0,2]$ denotes the fractional damping exponent and $\beta\in[0,1]$ the memory parameter. After showing the existence of a strong global attractor, we prove its upper semicontinuity in the parameter pair $(\alpha,\beta)$. We then construct a family of strong exponential attractors and establish its continuity in $(\alpha,\beta)$. Here, “strong" means that the compactness, attraction, and parameter-robustness properties are established in a topology stronger than that of the phase space. Compared to existing $\beta = 0$ results, our findings hold in a stronger topology. The analysis draws upon our recent higher-order regularity results for global attractors.

The classical Boltzmann equation, formulated by Boltzmann in 1872, describes the statistical behavior of a dilute gas in a non-equilibrium state. While the global well-posedness of the classical Boltzmann equation has been extensively studied and is relatively well-understood, this model becomes inadequate when the gas is no longer sufficiently dilute. Therefore, it is necessary to investigate higher-order extensions of the Boltzmann equation that account for multi-particle interactions. Recently, Ampatzoglou and Pavlović provided the rigorous derivation of both the ternary [Comm. Math. Phys. 387 (2021), no. 2, 793-863.] and the binary-ternary [Forum Math. Sigma 13 (2025), Paper No. e52, 95 pp.] Boltzmann equations from a classical particle system, yielding an explicit form of the ternary collision operator. However, the global existence near equilibrium for the binary-ternary Boltzmann equation remains an interesting and fundamental question, a gap that this paper seeks to address as its primary contribution.

We study ground states of the following Dirac-Coulomb system $$\mathscr{D}_cu_c- \left(\abs{x}^{-1}\ast \abs{u_c}^2\right)u_c=\omega_cu_c,$$ under the constraint $\norm[L^2]{u_c}^2=1.$For sufficiently large \(c\), we prove uniqueness of ground states modulo translations, phase transformations and spinorial rotations, and determine the kernel of the linearized operator at ground state. We also obtain asymptotic expansions of ground state in the nonrelativistic limit regime.

We study finite-time singularity formation for a nonlocal Boussinesq system arising in cold-plasma dynamics. We show that smooth, symmetric initial data with strictly positive total density and a sufficiently negative velocity slope at the origin lead to finite-time blow-up. More precisely, we give an explicit upper bound for the lifespan and show that the velocity gradient becomes unbounded.

We study a fractional Brezis–Nirenberg problem on a bounded domain with finitely many Hardy potentials centered at distinct points $a_1,\dots,a_k$, with masses $\mu_1,\dots,\mu_k\geq 0$. We assume $\sum_{i=1}^k\mu_i<\Lambda_{N,s}$ and consider $N>4s$ and $0<\mu_{\max}<\overline{\mu}_{N,s}$. The interaction between the poles allows existence for $\lambda=0$ and for some negative values of $\lambda$. At a pole $a_j$, this interaction is described by $B_j(\lambda,\mathbf a,\boldsymbol\mu)=\lambda+\sum_{i\neq j}\frac{\mu_i}{|a_i-a_j|^{2s}}$. This leads to a threshold $\lambda_{\mathrm{geo}}\leq 0$. We prove that the critical Rayleigh quotient is attained for $\lambda_{\mathrm{geo}}<\lambda<\lambda_1(\boldsymbol\mu,\mathbf a)$, and hence the problem admits a positive least-energy solution. If at least two masses are positive, then $\lambda_{\mathrm{geo}}<0$, so the result includes $\lambda=0$ and a nonempty interval of negative values of $\lambda$. We also prove a direct compactness estimate for weakly vanishing sequences by localizing the Gagliardo seminorm. The argument avoids both profile decomposition and the Caffarelli-Silvestre extension.

At an $O(2)$-equivariant Hopf bifurcation, the real parts $\zeta_R$ and $\xi_R$ of the two cubic normal-form coefficients govern selection between traveling and standing waves. In a fourth-order regularized $p$-system, $\xi_R$ was observed to vanish and the cubic nonlinearity to shift only frequencies. We show that both observations are structural: variational nonlinearities enforce them even when the linear part is dissipative and non-Hamiltonian. We consider a class of two-component PDEs on the circle whose quadratic and cubic nonlinearities are generated by reflection-invariant local Hamiltonians of the fields and finitely many of their spatial derivatives, through a common constant-coefficient Poisson operator. Then $\xi_R=0$ for every admissible linear part, while $\zeta_R$ depends only on the quadratic nonlinearity. For variational nonlinearities the direct cubic terms are purely imaginary. The mixed cascade term loses its real part because its second harmonic has zero temporal frequency and the resolvent keeps its parity, while the self-cascade at the doubled critical frequency keeps a real part. When $\zeta_R\ne0$, the bifurcating traveling waves are saddles, while the standing waves are supercritical and orbitally asymptotically stable for $\zeta_R<0$ and subcritical and unstable for $\zeta_R>0$. When the nonlinearities are the Poisson operator applied to polynomials in the fields alone, and its symbol does not vanish at the critical and second-harmonic wavenumbers, cancellation for every admissible linear part conversely characterizes variationality. In a dissipatively regularized Boussinesq family, as dissipation tends to zero, $\zeta_R$ vanishes linearly off resonance and diverges inversely at the 2:1 resonance. The Hamiltonian limit of the cubic coefficient is therefore singular.

This paper investigates the continuous dependence (stability) of solutions to the barotropic compressible Navier–Stokes equations in critical Besov spaces. Using the Lagrangian approach developed in \cite{Danchin2014}, it was shown that, for \(1<p<2d\), the flow map $(a_0,u_0)\mapsto (\bar a,\bar u)=(a\circ X,u\circ X)$ is Lipschitz continuous from $\dot B_{p,1}^{d/p}\times \dot B_{p,1}^{d/p-1}$ into $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$. However, this result does not directly imply the corresponding continuous dependence in the original Eulerian coordinates because of the low regularity of the critical initial data. Previously, only for $p<d$, continuous dependence was known only in certain lower-regularity spaces with a loss of one derivative relative to the natural solution spaces arising essentially as a by-product of the uniqueness argument. We close this gap and prove that for $1<p<2d$ the flow map $(a_0,u_0)\mapsto(a,u)$ is continuous (not Lipschitz continuous) from $\dot B_{p,1}^{d/p}\times\dot B_{p,1}^{d/p-1}$ to $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$ in Eulerian coordinates without any loss of regularity, which together with the known existence and uniqueness theory \cite{Danchin2014} completes Hadamard well-posedness in the critical spaces.

We study the diffusion limit of the classical solution to the relativistic Boltzmann equation with the initial data near a global relativistic Maxwellian. By using spectral analysis, we establish the convergence of the classical solution to the relativistic Boltzmann equation to that of the incompressible Navier-Stokes system, and give for the first time the optimal convergence rate and a precise estimate of the initial layer. Moreover, through a refined analysis of the linear collision operator $L$, combined with Duhamel's principle, we obtain the existence and uniqueness of a global strong solution to the relativistic Boltzmann equation, as well as a convergence rate independent of the speed of light $\mathbf{c}.$

This paper establishes the global existence of classical solutions to the initial-boundary value problem for the planar four-velocity Broadwell model in a multidimensional setting. While uniqueness was already established in our previous work (which addressed existence under the assumption of sufficiently small, bounded, and $C^{1}$-regular initial and boundary data), the objective of the present study is to completely lift this size restriction. By extending the previously developed fixed-point framework, we prove that the methodology remains highly effective for arbitrary initial and boundary data. Within this generalized setting, both the solutions and their first-order partial derivatives are shown to be well-defined and to exist globally for all times.

We prove a Pólya–Szegö inequality for functions defined on an $n$-dimensional submanifold $\Sigma$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $\Sigma$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo–Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser–Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya–Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $\Omega\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{\psi(s),\frac85\right\}g, \text{ where } \psi(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $\lambda_2-\lambda_1>46/5$.

We study the sharp $L^2$-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp $L^2$-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of $L^2$-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of $L^2$-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps $\mu_j-\mu_{j-1}$, and the coefficients of the second sum are $\mu_1$ times the gaps of the reciprocal spectrum, $\mu_1\left(\mu_{j-1}^{-1}-\mu_j^{-1}\right)$. On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when $N=2,3$ and by the degree three modes when $N\geq4$. In the Gaussian case, all the constants are rational numbers and the $L^2$-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.

In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \[\|\varphi\|_{L^p(v,\Omega)} \leq S(p,1) \| \sqrt{Q}\nabla \varphi\|_{L^p(\Omega)},\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\nabla u)\in QH^{1,p}_0(v,\Omega)$ to a Dirichlet problem for the $p$-Laplacian with a zero order term: \begin{equation*} -v^{-1}\text{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)+F|u|^{p-2}u = |f|^{p-2}f - v^{-1}\text{Div}(v|g|^{p-2}g\mathbf{t}),\; x \in \Omega, \quad \text{and} \quad u =0, \; x \in \partial \Omega, \end{equation*} More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.

This paper is concerned with the Fleming-Viot particle system introduced by Burdzy, Hołyst and March (2000). In this model, $N$ Brownian particles evolve independently in a bounded domain $D$ until one of the particles reaches the boundary. Then, the particle which has hit the boundary instantaneously jumps to the location of one of the other particles, chosen uniformly at random. Burdzy, Hołyst and March showed that when $N$ tends to infinity, the empirical measure of the system converges to a solution to the heat equation on $D$ with Dirichlet boundary conditions, renormalized to have total mass $1$. Our main result is a sharp, uniform-in-time, quantitative version of this result. We employ the method of weak propagation of chaos, which necessitates a careful study of the (backward) Kolmogorov equations associated to the $N$-particle systems, and the infinite-dimensional transport equation whose characteristics are given by renormalized solutions of the Dirichlet heat equation. The proofs are entirely analytical, and most of the technical effort is devoted to building barrier functions which are used to control the singular behavior of the system when most of the particles approach the boundary.