Papers

We study the long-time dynamics near Couette flow for the 2D Euler equations in unbounded geometries. We identify a mechanism of nonlinear inviscid damping at Yudovich regularity, distinct from classical phase mixing: velocity decay driven by spatial evacuation of vorticity. The mechanism leads to a geometry-sign classification of the dynamics. In the infinite channel, small nonnegative bounded-vorticity perturbations undergo global damping without derivative assumptions. In the whole plane, small nonnegative perturbations exhibit enhanced dispersion and velocity damping, while non-positive perturbations remain confined and do not damp, even when arbitrarily small in Gevrey classes. Moreover, damping can coexist with infinite-time growth of vorticity derivatives, even under arbitrary shear modulation. The classification is obtained through an interplay between new Lyapunov functionals and Hamiltonian conservation.

We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevskiĭ.

We study a parabolic transmission problem whose interface condition depends on the time derivative of the solution. This model arises naturally as the linearized limiting profile of a two-phase inhomogeneous Stefan problem after the hodograph transform. Our main result shows the existence and uniqueness of a classical solution. We also prove a Harnack inequality for a general class of transmission problems and establish $C^{1,\alpha}$ estimates up to the interface from both sides. These results can be used to obtain $C^{1,\alpha}$ regularity of flat free boundaries for the two-phase Stefan problem with distributed sources.

We study a thermodynamically consistent bulk-surface Navier–Stokes–Cahn–Hilliard system describing two-phase flows exhibiting moving contact lines, variable contact angles, and mass transfer. We establish the existence of global weak solutions for non-degenerate mobility functions and singular free-energy potentials. The proof is based on an implicit time-discretization scheme and a subdifferential characterization of the convex part of the bulk-surface free energy. Finally, we study the simultaneous high-friction and decoupling limit, and prove that corresponding weak solutions converge, up to a subsequence, to a weak solution of the Abels–Garcke–Grün model. One key ingredient of the proof is the Mosco convergence of the convex part of the bulk-surface free energy, which might be of independent interest.

We consider convex admissible viscosity solutions of $\sigma_2(D^2u)=f>0$ in an open subset of $\mathbb R^n$, where $n\ge2$, $0<\alpha<1$, and $f\in C_{\mathrm{loc}}^{0,\alpha}$. We prove that every such solution belongs to $C_{\mathrm{loc}}^{2,\alpha}$. For solutions in $B_2$, we also prove a uniform $C^{2,\alpha}(B_{1/4})$ estimate under an $L^\infty$ bound for $u$, a positive lower bound for $f$, and a $C^{0,\alpha}$ bound for $f$.

This paper concerns the shock formation problem for the 3D steady supersonic potential equation of polytropic gases. The potential equation is described by a second order quasilinear wave equation $\displaystyle\sum_{i=1}^{3}\big[(\partial_i\Phi)^2 - c^2(\rho)\big]\partial_i^2\Phi + 2\displaystyle\sum_{1\le i<j\le 3}\partial_i\Phi\partial_j\Phi\partial_{ij}^2\Phi = 0$, where $x = (x^1,x^2,x^3)$, $(\partial_1,\partial_2,\partial_3)=(\partial_{x^1},\partial_{x^2},\partial_{x^3})$, $c(\rho) = \sqrt{p'(\rho)}$ is the sonic speed with $p(\rho)=A\rho^{\gamma}$ ($\gamma>1$), and $\partial_3\Phi > c(\rho)$. For the short pulse boundary data $\Phi|_{x^3=0} = \delta^\nu\Phi_0\big(\frac{r-1}{\delta},\omega\big)$ and $\partial_3\Phi|_{x^3=0}=q_0+\delta^{\nu-1}\Phi_1\big(\frac{r-1}{\delta},\omega\big)$ with $r=\sqrt{(x^1)^2+(x^2)^2}$, $\omega=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\in\mathbb{S}$, $1<\nu<2$ and small $\delta>0$, it is shown that a shock will be formed in a finite $x^3$-distance as long as the boundary data are supersonic and satisfy $(\Phi_0,\Phi_1)\not\equiv 0$. This coincides with physical phenomenon that strong compression of supersonic polytropic gases yields shocks. One of our main ingredients is to find a good unknown so that the previously imposed compatibility conditions on the short pulse initial data are removed as well as the required weighted energy estimates in the existing literatures are derived. It is expected that the method here will be applied to study the shock formation problem with general short pulse initial data for the 3D steady supersonic Euler equations of polytropic gases.

We study spectral and variational properties of the (possibly) fractional Dirichlet Laplacian $\left(-\Delta_{n+k}\right)^s\!$, $s>0$, on bounded subsets of $\mathbb R^n\times\mathbb R^k$ whose extent in one or more directions becomes much larger than in the remaining ones. We first investigate the asymptotic behaviour of the first eigenvalue on stretching (or thinning) tubes, covering in particular the case of integers $s\geq 2$, for which $\left(-\Delta_{n+k}\right)^s\!$ reduces to a polyharmonic operator. Then we compute the $G$-limit of the rescaled operators, and the $\Gamma$-limit of the corresponding rescaled quadratic forms.

Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen–Jian–Zhou with the algebraic reduction of the quotient equation to a $\sigma _2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(\Delta u+a)$. We work with the linearized operator $G=(\Delta u-f)I-D^2u$ of the equivalent equation $\sigma_2(D^2u)=f \Delta u$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(\Delta f\) term by integration by parts solely in terms of the Lipschitz norm of \(f\). A Legendre–Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.

Let $\Omega\subset\mathbb{R}^d$ be a bounded smooth domain and let $-\Delta_D$ be the positive Dirichlet Laplacian on $L^2(\Omega)$. For a real polynomial $p$ with positive leading coefficient, we study the constrained linear equation \[ u_t=-\Pi_u\,p(-\Delta_D)u, \qquad \lVert u(0)\rVert_{L^2}=1. \] Its solution is the normalized semigroup orbit \[ u(t)=\frac{e^{-t p(-\Delta_D)}u_0} {\lVert e^{-t p(-\Delta_D)}u_0\rVert_{L^2}}. \] The active spectral support is preserved, and the trajectory converges to the normalized projection of $u_0$ onto the active eigenspaces for which $p(\lambda_j)$ is minimal. The next active polynomial spectral value gives the exponential rate. For every $\theta\geq 0$ and $\tau>0$, the same rate holds in the domain of $(-\Delta_D)^\theta$ for $t\geq\tau$, even when the initial datum has no fractional regularity. We also show that a finite set of Dirichlet levels can be prescribed as the global minimizing set of a polynomial, and that an isolated selected set is stable under sufficiently small polynomial perturbations. For $p(s)=s^m$, the lowest active Dirichlet level is selected. For $p(s)=(s-\rho)^2$, selection is by distance from $\rho$, and cross-level degeneracy occurs only at Dirichlet midpoints.

In this paper we study nonlocal nonlinear equations of $s$-fractional $p$-Laplacian type in open subsets of $\mathbf{R}^n$. We investigate how the boundary regularity of solutions depends on the parameters $s$ and $p$, including the local case $s=1$. Specifically, we show exactly when regularity for $(s_1, p_1)$ implies regularity for $(s_2, p_2)$. The proof relies on the equivalence between Wiener criteria formulated with condenser and Sobolev capacities. To establish this equivalence, we derive precise comparison estimates between the two capacities. The Wiener integral defines thinness and the fine topology. We show that every superharmonic function associated with a nonlocal nonlinear operator is finely continuous. Moreover, we prove that polar sets in this fractional setting coincide with sets of zero capacity.

Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $\Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=\Omega\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $\Omega \in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.

We give a roadmap for the study of critical fractional Hardy inequalities for Dirichlet and Sobolev–Bregman forms. We focus on homogeneous forms on the real line, in particular the form of the fractional Laplacian and the Servadei–Valdinoci form. We prove criticality results in both the Dirichlet and Sobolev–Bregman settings.

In this manuscript, we are concerned with the hierarchical control of a non-degenerate quasi-linear parabolic system, posed on a bounded interval. We use the Stackelberg-Nash strategy with a single control called leader and two controls called followers. First, we study the controllability for the linear system, where we will demonstrate a Carleman inequality associated with the adjoint problem. Then, using the results obtained for the linear case and applying Liusternik's Inverse Function Theorem, we obtain local null controllability for the quasi-linear problem.

At an $O(2)$-equivariant Hopf bifurcation a spatially periodic system selects between traveling waves and standing waves, and which branch appears and whether it is stable is decided by two cubic normal form coefficients. Obtaining these coefficients for a given PDE has required a derivation carried out afresh for each model. We remove that step for two-component systems with conservative polynomial differential nonlinearities on a one-dimensional periodic domain, deriving closed form formulas for both coefficients. They are expressed directly in terms of the array of nonlinear PDE coefficients and the spectral data of the linearization, its critical eigenvectors, adjoint eigenvectors, and non-critical resolvents. We give verifiable conditions under which the underlying center manifold reduction is valid: a structural condition on the principal part of the linearization that yields the required resolvent estimate, and a condition on the Fourier symbol that we show is equivalent to the required spectral gap. In contrast to the model specific computations available previously, the resulting formulas apply to any system in the class without further derivation. A companion implementation evaluates the two normal form coefficients from the coefficient array and verifies the assumptions for a given system. We specialize the formulas to two applications: a strain-gradient regularization of the nonlinear $p$-system, whose quadratic case contains a previously studied model as a special case, and a first order conservative bilinear family, not previously analyzed, in which every bifurcation scenario allowed by the general classification is realized.

We give a short proof of global regularity for the critical dissipative SQG equation in two dimensions. The argument is based on a pointwise representation of the Leibniz commutator for the half-Laplacian, recently obtained by the author with Luca Gennaioli, which simplifies and unifies the nonlinear estimates used in previous maximum principle proofs.

In this paper, we propose a novel physical model for traffic flow incorporating nonlinear anticipation effects and an intelligent-control boundary, mathematically formulated as a damping boundary condition: \begin{align*} \begin{cases} u_t^\tau+v_x^\tau=0, & x\in(0,1),\; t>0,\\[1mm] v_t^\tau+g(u_x^\tau)u_x^\tau=\dfrac{f(u^\tau)-v^\tau}{\tau}, & x\in(0,1),\; t>0,\\[1mm] (u^\tau,v^\tau)(x,0)=(u_0^\tau(x),v_0^\tau(x)), & x\in(0,1),\\[1mm] u_x^\tau(0,t)=0,\quad u_x^\tau(1,t)=-ku_t^\tau(1,t), & k>0,\; t>0, \end{cases} \end{align*} where $f$ and $g$ are smooth functions satisfying suitable structural assumptions, and $\tau>0$ is the relaxation time. The primary objective is to rigorously investigate how the intelligent-control boundary suppresses the stop-and-go phenomenon in the large-relaxation-time regime-a mechanism that has not been mathematically addressed in previous studies. Utilizing the energy method, we establish the global well-posedness and exponential time-decay of solutions for the original system under arbitrarily large initial data with small spatial derivatives. Furthermore, we analyze the asymptotic behavior in the large-relaxation-time limit $\tau\to\infty$, by introducing a novel technique that incorporates constant shifts into the initial data of the limiting system. This approach enables us to construct a modified auxiliary system, through which we successfully obtain the global convergence of the original solutions to the asymptotic profiles for all time $t$ as relaxation time $\tau\to\infty$. Numerical simulations further demonstrate that in the large-relaxation-time regime, the stop-and-go density waves emerging at the early stage are gradually suppressed by the damping boundary. This leads the traffic stream to eventually evolve into an essentially uniform profile, which perfectly validates our theoretical results.

We study a broad family of active scalar equations with fractional dissipation on $\mathbb{R}^2$ and prove that their entropy diverges to $-\infty$ at a sharp rate of order $\log t$. The proof is based on a suitable regularization scheme, uniform entropy estimates, and a limiting argument via the Vitali convergence theorem. Our analysis introduces new fractional logarithmic Sobolev inequalities, weighted commutator estimates, and fractional moment bounds. These results provide a general framework for studying spatial decay and long-time dynamics of nonlocal nonlinear partial differential equations.

We study three-dimensional wave scattering by high-index dielectric resonators with Kerr-type nonlinearity under plane-wave incidence, together with the associated nonlinear dielectric scattering resonances, through a nonlinear Lippmann-Schwinger equation. Using a Lyapunov-Schmidt reduction near a simple eigenmode of the Newtonian potential, we obtain a local decomposition of the scattered wave into a resonant contribution and a controlled remainder and derive an explicit nonlinear equation for the resonant modal coefficient for sufficiently small contrast parameter and locally small resonant amplitude and incident field. A second reduction covers the regime of incident fields of order one and weak nonlinearity. Our results extend for the first time the linear modal decomposition to nonlinear wave-scattering problems. We complement them by developing a numerical framework based on Nyström discretization, real Newton iteration, and pseudo-arclength continuation. For single resonators of several geometries, our computations confirm the predicted high-contrast resonance scaling and the nonlinear modal approximation, and exhibit the multivalued incident-wave response. For a mirror-symmetric resonator dimer, we track symmetric, antisymmetric, and symmetry-broken resonance branches over a broad range of separations. Our computations show that the symmetry-breaking threshold increases as the gap between the resonators decreases, in agreement with the leading-order bifurcation theory.

We study abstract wave equations with time-dependent propagation speed under strict hyperbolicity. Our starting point is the observation that a substantial part of the classical energy theory, developed through different constructions, can be organized around a common variational mechanism. Approximate energies naturally lead to regularity–fidelity problems in which a smooth approximation of the reciprocal propagation speed is chosen by balancing derivative size against distance from the original coefficient. We combine this approximation mechanism with higher-order energy corrections and obtain a hierarchy of variational problems together with a cascade of almost conserved energies of arbitrary order. The resulting variational problems are then studied independently of the evolution equation. A qualitative distinction emerges between the first and higher orders: for arbitrary prescribed growth, all higher orders generate the same classes, while the first-order problem may behave differently. We also establish connections with fractional and generalized fractional Sobolev regularity. This framework provides a unified interpretation of several classical sufficient conditions for energy control and derivative-loss estimates. At the same time, it identifies a natural limitation of the traditional approach, namely the use of absolute integrability of energy errors. Once this limitation is recognized, a different regime becomes accessible: by retaining their oscillatory structure and exploiting cancellations, we construct strictly hyperbolic propagation speeds beyond the regularity thresholds detected by the variational theory, for which uniform energy estimates and hence no derivative loss nevertheless hold. Thus the variational viewpoint reveals an unexpected common structure behind a substantial part of the classical theory and indicates a genuinely different regime beyond it.

We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^\alpha u=Lu-\lambda u+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, $\partial_t^\alpha$ is the Caputo derivative of order $\alpha\in(0,1)$, and $L$ is a spatially nonlocal operator of order $\sigma\in(0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $\alpha\to1$ and $\sigma\to2$. We also establish unique solvability when either $T<\infty$ or $\lambda>0$. The proof is based on a direction-by-direction extension argument.

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

We study the rigidity of traveling waves and long-time species selection in a one-dimensional critical three-species competition-diffusion system. We establish several rigidity results for traveling waves connecting distinct equilibria on the critical simplex. In the case where one species has a strictly larger Fisher-KPP spreading speed than the other two, an entropy method, combined with Gagliardo-Nirenberg and Nash inequalities yieldsuniform extinction of the slower species and convergence to the fastest-species equilibrium throughout every cone with speed below its KPP speed.

We construct a plane Lipschitz domain for which the range of the Laplace operator from $H^{3/2}\cap H^1_0$ to $H^{-1/2}$ is not closed. We show that rapid oscillations of the boundary create a boundary layer that leads to unbounded $H^{3/2}$ norm of the solution of the Dirichlet problem. Our proof is elementary in the sense that it is directly based on homogeneity arguments and does not use deep results of harmonic analysis such as estimates of area integrals or harmonic measures of Dahlberg or Jerison and Kenig.

This paper develops a fully discrete finite element method for three-dimensional incompressible magnetohydrodynamic (MHD) flows with nonlinear $p$-Laplace viscosity. The scheme combines spatial finite elements with a semi-implicit Euler time discretisation. Convergence to a weak solution is proved using a time-translation compactness argument together with Minty's monotonicity method. Under additional regularity assumptions, we derive unconditional error estimates for both the velocity and magnetic field, with no coupling restriction between the time step and mesh size. The framework extends finite element analysis of incompressible MHD systems to non-Newtonian shear-thickening fluids ($p>2$), while recovering the classical Newtonian case ($p=2$). Finally, numerical simulations are provided to validate the theoretical convergence rates and demonstrate the robustness of the proposed method.

Motivated by dissipative approximations of the three-dimensional focusing energy-critical nonlinear Schrödinger equation, we study the inviscid and zero-dispersion limits for general energy-critical complex Ginzburg-Landau equations in dimensions $3\le d\le6$. We develop a new framework for these limits that does not require a global theory for the limiting equation. For limiting data in $H^1$, we prove strong convergence in the critical space, in particular in $L_t^\infty H_x^1$, on every compact subinterval of the maximal lifespan of the limiting solution. This includes strong $H^1$ inviscid convergence to the 3D focusing energy-critical NLS without higher regularity of the limiting data. The diffusion and nonlinear coefficients may vary independently, and both focusing and defocusing cases are treated. Both limits use coefficient-uniform critical stability. The inviscid argument uses low-frequency truncation to overcome the derivative loss at $H^1$, whereas uniform parabolic smoothing yields a quantitative $H^1$ estimate in the zero-dispersion limit. Moreover, for the inviscid limit, we prove the required homogeneous Strichartz estimates and obtain the retarded estimates by establishing a uniform retarded double-endpoint bound for the critical forcing space.

We study the vanishing-regularization limit of the regularized subjective-surface model introduced for touching and dividing cell nuclei and previously analyzed for fixed positive regularization parameters. On a smooth bounded domain $\Omega\subset\mathbb R^d$, the model is \[ \begin{aligned} \partial_tu^{\eps,\nu} ={}&\nu\Delta u^{\eps,\nu} +A_\eps(\nabla u^{\eps,\nu}) \operatorname{div}\!\left( G(x)\frac{\nabla u^{\eps,\nu}} {A_\eps(\nabla u^{\eps,\nu})} \right) -\mu R_\eta(x,u^{\eps,\nu}),\\ &A_\eps(p)=\sqrt{\eps^2+|p|^2}, \qquad R_\eta(x,r)=\Lambda(x)H_\eta(r-q), \end{aligned} \] subject to homogeneous Dirichlet data and a prescribed initial profile. We formulate the limiting weighted level-set mean-curvature equation as a viscosity initial-boundary value problem and establish global existence, uniqueness, preservation of $[0,1]$, and nonexpansive dependence on the initial data. We determine the direct upper and lower limiting values of the regularized principal operators at zero gradient and identify the additional argument needed to recover the geometric directional values. For every $T>0$, the regularized solutions converge uniformly to the unique viscosity solution on $[0,T]\times\overline\Omega$ as $\eps,\nu\to0$, independently of their relative rates of decay. This determines the geometric limit of the regularized model used in 3D and 3D+time microscopy segmentation.

In this paper, we study optimal control problems for stochastic semilinear partial differential equations, which lack the maximum principle, and whose coefficients do not have bounded Frechet derivatives. We propose an approximation scheme for the corresponding optimization problem, and prove convergence of the approximating solutions on both finite and infinite time intervals.

Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.