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Andrea Malchiodi

Publications and source records attributed to Andrea Malchiodi.

At least 19 recordsLinked to original sources
arXiv2026

Quantization for Palais-Smale sequences of the Liouville functional on closed surfaces

In this paper we prove a quantization property for Palais-Smale sequences of functionals related to Liouville equations on compact surfaces. While these equations have been extensively studied over the past decades, such a quantization property had not been established so far, and has been often bypassed by the Struwe monotonicity trick. Our result, which relies on careful $L^p$ estimates to obtain gradient bounds and to perform neck analysis, allows to directly apply variational methods to obtain existence of solutions in non-resonant regimes. We also give an example of clustering of blow-up points and a stronger assumption that, on the contrary, guarantees that the blow-up points are isolated.

math.AP↗
arXiv2026

Decreasing Weyl's energy by connected sums with locally conformally flat manifolds

We study the Weyl functional on connected sums of two four-dimensional manifolds $(M,g_M)$ and $(Z,g_Z)$, assuming $g_M$ is Bach-flat and $g_Z$ locally conformally flat. We show that if $g_M$ is neither self-dual nor anti self-dual and if $g_Z$ is of positive Yamabe class, there exists a metric $g_Y$ on $Y := M \# Z$ with Weyl energy lower than that of $g_M$ (with the trivial exception of $(Z,g_Z) = (\mathbb{S}^4, g_{\mathbb{S}^4})$). This result has a relation to a conjecture by Singer and has a perspective application to the minimization of Weyl's energy. The proof relies on a simultaneous interplay of $W_M^+, W_M^-$ and the topology of $Z$, and also covers some orbifold cases.

math.DG↗
arXiv2025

Mass and volume of four-dimensional Einstein metrics

Let $(M^4,\bar{g})$ be an Einstein manifold, where $M^4$ is a smooth, closed, oriented four-manifold $M^4$ and $\bar{g}$ has positive Einstein constant. Given a point $0 \in M^4$, let $G$ denote the (positive) Green's function $G$ of the conformal laplacian $L_{\bar{g}}$; then $g = G^2 \bar{g}$ is a complete, scalar-flat, asymptotically flat metric on $\widehat{M} = M \setminus \{ 0 \}$. We first show that the ADM mass of $g$ can be expressed as an integral over $\widehat{M}$, then use this identity to prove a lower bound for the mass of $g$ in terms of the volume of $\bar{g}$. As corollaries, we prove a 'mass times volume' inequality, plus various mass gap theorems characterizing the round metric on $S^4$ and the Fubini-Study metric on $\mathbb{CP}^2$.

math.DG↗
arXiv2025

Min-max theory and Yamabe metrics on conical four-manifolds

We prove existence of Yamabe metrics on four-manifolds possessing finitely-many conical points with $\mathbb{Z}_2$-group, using for the first time a min-max scheme in the singular setting. In our variational argument we need to deform continuously regular bubbles into singular ones, while keeping the Yamabe energy sufficiently low. For doing this, we exploit recent positive mass theorems in the conical setting and study how the mass of the conformal blow-up diverges as the blow-up point approaches the singular set.

math.DG↗
arXiv2025

Weyl energy and connected sums of four-manifolds

Given two closed, oriented Riemannian four-manifolds $(M,g_M)$ and $(Z,g_Z)$, which are not locally conformally flat and not both self-dual or both anti-self-dual, we prove that there exists a metric $g_Y$ on the connected sum $Y\cong M\#Z$ such that the Weyl energy of $g_Y$ is strictly smaller than the sum of Weyl energies of $g_M$ and $g_Z$.

math.DG↗
arXiv2024

Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

math.AP↗
arXiv2024

Sharp bounds on the Nusselt number in Rayleigh-Bénard convection and a bilinear estimate via Carleson measures

We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit. Due to a maximum principle property for the temperature exploited by Constantin-Doering and Otto-Seis, this amounts to proving a-priori bounds for horizontally-periodic solutions of a fourth-order equation in a strip of large width. Such bounds are obtained here using Fourier analysis, integral representations, and a bilinear estimate due to Coifman and Meyer which uses the Carleson measure characterization of BMO functions by Fefferman.

math.AP↗
arXiv2024

Yamabe metrics on conical manifolds

We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin's classical result. Interestingly, the singular nature of the metric determines a different condition on the dimension, compared to the regular case. We derive asymptotic expansions on the Yamabe quotient by adding a proper and implicit lower-order correction to standard bubbles, whose contribution to the expansion of the quotient can be determined combining the decomposition of symmetric two-tensor fields and Fourier analysis on the conical links.

math.DG↗
arXiv2023

Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model

Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action $$ I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle $$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail.

math.AP↗
arXiv2023

A factorization of the GJMS operators of special Einstein products and applications

We show that the GJMS operators of a special Einstein product factor as a composition of second- and fourth-order differential operators. In particular, our formula applies to the Riemannian product $H^{\ell} \times S^{d-\ell}$. We also show that there is an integer $D = D(k,\ell)$ such that if $d \geq D$, then for any special Einstein product $N^\ell \times M^{d-\ell}$, the Green's function for the GJMS operator of order $2k$ is positive. As a result, these products give new examples of closed Riemannian manifolds for which the $Q_{2k}$-Yamabe problem is solvable.

math.DG↗
arXiv2023

On the variation of the Einstein-Hilbert action in pseudohermitian geometry

In this paper we compute the first and second variation of the normalized Einstein-Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater or equal to five, in three dimension we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

math.DG↗
arXiv2023

Łojasiewicz inequalities near simple bubble trees

In this paper we prove a gap phenomenon for critical points of the $H$-functional on closed non-spherical surfaces when $H$ is constant, and in this setting furthermore prove that sequences of almost critical points satisfy Łojasiewicz inequalities as they approach the first non-trivial bubble tree. To prove these results we derive sufficient conditions for Łojasiewicz inequalities to hold near a finite-dimensional submanifold of almost-critical points for suitable functionals on a Hilbert space.

math.AP↗
arXiv2022

Critical points of the Moser-Trudinger functional on closed surfaces

Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathcal{E}_β:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=β\right\}$ for any $β>0$.

math.AP↗
arXiv2022

A note on the critical Laplace Equation and Ricci curvature

We study strictly positive solutions to the critical Laplace equation \[ - Δu = n(n-2) u^{\frac{n+2}{n-2}}, \] decaying at most like $d(o, x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative Ricci curvature, of dimension $n \geq 3$. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless $(M, g)$ is isometric to $\mathbb{R}^n$ and $u$ is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of $u$.

math.AP↗
arXiv2021

Double Bubbles with High Constant Mean Curvatures in Riemannian Manifolds

We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.

math.DG↗