arXiv subjects

Tobias König

Publications and source records attributed to Tobias König.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Yamabe stability optimizers

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin--Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

math.DG↗

arXiv2026

An almost-almost-Schur lemma on the 3-sphere

In the conformal class of the standard metric on the $3$-sphere, we prove a quantitative refinement of the Andrews-De Lellis-Topping inequality in terms of a two-term distance to the set of minimizing conformal factors. This inequality is itself a stability result for the well-known Schur lemma and is therefore referred to as almost-Schur lemma. Hence, our stability result may be viewed as an almost-almost-Schur lemma. As a consequence, we deduce via interpolation the quantitative stability of an entire family of nonlinear Yamabe-type inequalities, including an inequality for the total volume-normalized $σ_2$-curvature $\mathcal F_2$. This extends a recent result by Frank and the second author for $d > 4$ to the case $d=3$. While the standard metric minimizes $\mathcal F_2$ if $d > 4$, it maximizes $\mathcal F_2$ if $d=3$. This is the main challenge in treating the case $d=3$ as it turns the related functional inequality into a reverse Sobolev-type inequality.

math.AP↗

arXiv2025

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

For a smooth bounded domain $Ω\subset \mathbb R^3$ and smooth functions $a$ and $V$, we consider the asymptotic behavior of a sequence of positive solutions $u_ε$ to $-Δu_ε+ (a+εV) u_ε= u_ε^5$ on $Ω$ with zero Dirichlet boundary conditions, which blow up as $ε\to 0$. We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension $N=3$.

math.AP↗

arXiv2025

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

For a bounded set $Ω\subset \mathbb R^N$ and a perturbation $V \in C^1(\overlineΩ)$, we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+ εV = N(N-2) u_ε^\frac{N+2}{N-2} \] for dimensions $N \geq 4$, which are non-critical in the sense of the Brezis--Nirenberg problem. For the general case of multiple concentration points, we prove that concentration points are isolated and characterize the vector of these points as a critical point of a suitable function derived from the Green's function of $-Δ$ on $Ω$. Moreover, we give the leading order expression of the concentration speed. This paper, with a recent one by the authors (arXiv:2208.12337) in dimension $N = 3$, gives a complete picture of blow-up phenomena in the Brezis-Nirenberg framework.

math.AP↗

arXiv2025

Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere

We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

math.AP↗

arXiv2024

Sharp extinction rates for positive solutions of fast diffusion equations

Let $s \in (0, 1]$ and $N > 2s$. It is known that positive solutions to the (fractional) fast diffusion equation $\partial_t u + (-Δ)^s (u^\frac{N-2s}{N+2s}) = 0$ on $(0, \infty) \times \mathbb R^N$ with regular enough initial datum extinguish after some finite time $T_* > 0$. More precisely, one has $\frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 =o(1)$ as $t \to T_*^-$ for a certain extinction profile $U_{T_*, z, λ}$, uniformly on $\mathbb R^N$. In this paper, we prove the quantitative bound $ \frac{u(t,\cdot)}{U_{T_*, z, λ}(t,\cdot)} - 1 = \mathcal O( (T_*-t)^\frac{N+2s}{N-2s+2})$, in a natural weighted energy norm. The main point here is that the exponent $\frac{N+2s}{N-2s+2}$ is sharp. This is the analogue of a recent result by Bonforte and Figalli (CPAM, 2021) valid for $s = 1$ and bounded domains $Ω\subset \mathbb R^N$. Our result is new also in the local case $s = 1$. The main obstacle we overcome is the degeneracy of an associated linearized operator, which generically does not occur in the bounded domain setting. For a smooth bounded domain $Ω\subset \mathbb R^N$, we prove similar results for positive solutions to $\partial_t u + (-Δ)^s (u^m) = 0$ on $(0, \infty) \times Ω$ with Dirichlet boundary conditions when $s \in (0,1)$ and $m \in (\frac{N-2s}{N+2s}, 1)$, under a non-degeneracy assumption on the stationary solution. An important step here is to prove the convergence of the relative error, which is new for this case.

math.AP↗

arXiv2024

Quantitative stability of the total $Q$-curvature near minimizing metrics

Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k < \frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \leq k< \frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field.

math.AP↗

arXiv2024

Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality

The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y \geq Λ_{n, s, p, α,β} \|u |x|^{-β}\|_{L^p}^2, $$ for $0<s<\min\{1, n/2\}$, $2<p<2^*_s$, and $α,β\in\mathbb R$ so that $β-α= s - n\big(\frac12 - \frac1p\big)$ and $-2s < α< \frac{n-2s}{2}$. Continuing the program started in Ao et al. (2022), we establish the non-degeneracy and sharp quantitative stability of minimizers for $α\ge 0$. Furthermore, we show that minimizers remain symmetric when $α<0$ for $p$ very close to $2$. Our results fit into the more ambitious goal of understanding the symmetry region of the minimizers of the fractional Caffarelli-Kohn-Nirenberg inequality. We develop a general framework to deal with fractional inequalities in $\mathbb R^n$, striving to provide statements with a minimal set of assumptions. Along the way, we discover a Hardy-type inequality for a general class of radial weights that might be of independent interest.

math.AP↗

arXiv2024

An exceptional property of the one-dimensional Bianchi-Egnell inequality

In this paper, for $d \geq 1$ and $s \in (0,\frac{d}{2})$, we study the Bianchi-Egnell quotient \[ \mathcal Q(f) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B} \frac{\|(-Δ)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{\frac{2d}{d-2s}}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal B)^2}, \qquad f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B, \] where $S_{d,s}$ is the best Sobolev constant and $\mathcal B$ is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when $d = 1$, there is a neighborhood of $\mathcal B$ on which the quotient $\mathcal Q(f)$ is larger than the lowest value attainable by sequences converging to $\mathcal B$. This behavior is surprising because it is contrary to the situation in dimension $d \geq 2$ described recently in \cite{Koenig}. This leads us to conjecture that for $d = 1$, $\mathcal Q(f)$ has no minimizer on $\dot{H}^s(\mathbb R^d) \setminus \mathcal B$, which again would be contrary to the situation in $d \geq 2$. As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every $d \geq 1$. For $d \geq 2$, this family yields an alternative proof of the main result of \cite{Koenig}. For $d =1$ we make some numerical observations which support the conjecture stated above.

math.AP↗

arXiv2023

Stability for the Sobolev inequality: existence of a minimizer

We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\mathcal M$ and given by \[ \frac{\|\nabla f\|_{L^2(\mathbb R^d)}^2 - S_d \|f\|_{L^\frac{2d}{d-2}(\mathbb R^d)}^2}{ \inf_{h \in \mathcal M} \|\nabla (f - h)\|_{L^2(\mathbb R^d)}^2 } \geq c_{BE} > 0 \qquad \text{ for every } f \in \dot{H}^1(\mathbb R^d),\] which is due to Bianchi and Egnell, admits a minimizer for every $d \geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_{BE} < 2 - 2^\frac{d-2}{d}$, which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent $s \in (0, d/2)$ and dimension $d \geq 2$.

math.AP↗

arXiv2023

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in \dot H^s(\mathbb R^N)$ whose energy satisfies $$\tfrac{1}{2} S^\frac{N}{2s}_{N,s} \le \|u\|_{\dot H^s(\mathbb R^N)} \le \tfrac{3}{2}S_{N,s}^\frac{N}{2s},$$ where $S_{N,s}$ is the optimal Sobolev constant, the bound $$ \|u -U[z,λ]\|_{\dot{H}^s(\mathbb R^N)} \lesssim \|(-Δ)^s u - u^{2^*_s-1}\|_{\dot{H}^{-s}(\mathbb R^N)}, $$ holds for a suitable fractional Talenti bubble $U[z,λ]$. {For functions $u$ which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality.} As an application {of this}, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.

math.AP↗

arXiv2022

On the sharp constant in the Bianchi-Egnell stability inequality

This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s \in (0, \frac{d}{2})$. We prove that in dimension $d \geq 2$ the best constant \[ c_{BE}(s) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal M} \frac{\|(-Δ)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{2^*}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal M)^2} \] is strictly smaller than the spectral gap constant $\frac{4s}{d+2s+2}$ associated to sequences which converge to the manifold $\mathcal M$ of Sobolev optimizers. In particular, $c_{BE}(s)$ cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion of the Bianchi-Egnell quotient along a well-chosen sequence of test functions converging to $\mathcal M$.

math.AP↗

arXiv2022

Blow-up of solutions of critical elliptic equations in three dimensions

We describe the asymptotic behavior of positive solutions $u_ε$ of the equation $-Δu + au = 3\,u^{5-ε}$ in $Ω\subset\mathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_ε$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Δu + (a+εV) u = 3\,u^5$ in $Ω$.

math.AP↗

arXiv2022

Singular metrics of constant negative $Q$-curvature in Euclidean spaces

We study singular metrics of constant negative $Q$-curvature in the Euclidean space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to the problem \[ (-Δ)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash \{0\}, \] under a finite volume condition $Λ:=\int_{\mathbb{R}^n}e^{nu}dx$. We classify all singular solutions of the above equation based on their behavior at infinity and zero. As a consequence of this, when $n=1,2$, we show that there is actually no singular solution. Then adapting a variational technique, we obtain that for any $n\geq 3$ and $Λ>0$, the equation admits solutions with prescribed asymptotic behavior. These solutions correspond to metrics of constant negative $Q$-curvature, which are either smooth or have a singularity at the origin of logarithmic or polynomial type. The present paper complements previous works on the case of positive $Q$-curvature, and also sharpens previous results in the nonsingular negative $Q$-curvature case.

math.AP↗

arXiv2022

Constant $Q$-curvature metrics with a singularity

For dimensions $n \geq 3$, we classify singular solutions to the generalized Liouville equation $(-Δ)^{n/2} u = e^{nu}$ on $\mathbb{R}^n \setminus \{0\}$ with the finite integral condition $\int_{\mathbb{R}^n} e^{nu} < \infty$ in terms of their behavior at $0$ and $\infty$. These solutions correspond to metrics of constant $Q$-curvature which are singular in the origin. Conversely, we give an optimal existence result for radial solutions. This extends some recent results on solutions with singularities of logarithmic type to allow for singularities of arbitrary order. As a key tool to the existence result, we derive a new weighted Moser--Trudinger inequality for radial functions.

math.AP↗

arXiv2021

Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

For $s \in (0,1)$ and a bounded open set $Ω\subset \mathbb R^N$ with $N > 2s$, we study the fractional Brezis--Nirenberg type minimization problem of finding $$ S(a) := \inf \frac{\int_{\mathbb R^N} |(-Δ)^{s/2} u|^2 + \int_Ωa u^2}{\left( \int_Ωu^\frac{2N}{N-2s} \right)^\frac{N-2s}{N}}, $$ where the infimum is taken over all functions $u \in H^s(\mathbb R^N)$ that vanish outside $Ω$. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $N \in (2s, 4s)$, we prove that the Robin function $ϕ_a$ satisfies $\inf_{x \in Ω} ϕ_a(x) = 0$, which extends a result obtained by Druet for $s = 1$. In dimensions $N \in (8s/3, 4s)$, we then study the asymptotics of the fractional Brezis--Nirenberg energy $S(a + \varepsilon V)$ for some $V \in L^\infty(Ω)$ as $\varepsilon \to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points.

math.AP↗

arXiv2021

Reverse conformally invariant Sobolev inequalities on the sphere

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$-sphere involving an operator of order $2s> n$. In this case the Sobolev exponent is negative. Our results extend existing ones to noninteger values of $s$ and settle the question of validity of a corresponding inequality in all dimensions $n\geq 2$.

math.AP↗

arXiv2020

Classification of solutions of an equation related to a conformal log Sobolev inequality

We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to Beckner. Our proof uses an extension of the method of moving spheres from $\mathbb R^n$ to $\mathbb S^n$ and a classification result of Li and Zhu. Along the way we prove a small volume maximum principle and a strong maximum principle for the underlying operator which is closely related to the logarithmic Laplacian.

math.AP↗