Vol. 2024
|
Bortz, Simon; Poggi, Bruno; Tapiola, Olli; Tolsa, Xavier The A∞ condition, ε-approximators, and Varopoulos extensions in uniform domains To appear in The Journal of Geometric Analysis, 2024. Abstract | Links @unpublished{nokey,
title = {The A∞ condition, ε-approximators, and Varopoulos extensions in uniform domains },
author = { Simon Bortz and Bruno Poggi and Olli Tapiola and Xavier Tolsa},
url = {https://arxiv.org/abs/2302.13294},
year = {2024},
date = {2024-04-05},
urldate = {2023-02-26},
abstract = {Suppose that Ω⊂Rn+1, n≥1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω. We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂Ω in the sense that ωL∈A∞(σ) if and only if any bounded solution u to Lu=0 in Ω is ε-approximable for any ε∈(0,1). By ε-approximability of u we mean that there exists a function Φ=Φε such that ∥u−Φ∥L∞(Ω)≤ε∥u∥L∞(Ω) and the measure μ˜Φ with dμ˜=|∇Φ(Y)|dY is a Carleson measure with L∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis. },
howpublished = {To appear in The Journal of Geometric Analysis},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Suppose that Ω⊂Rn+1, n≥1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω. We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂Ω in the sense that ωL∈A∞(σ) if and only if any bounded solution u to Lu=0 in Ω is ε-approximable for any ε∈(0,1). By ε-approximability of u we mean that there exists a function Φ=Φε such that ∥u−Φ∥L∞(Ω)≤ε∥u∥L∞(Ω) and the measure μ˜Φ with dμ˜=|∇Φ(Y)|dY is a Carleson measure with L∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis. |
Bortz, Simon; Hofmann, Steve; Luna, José Luis; Mayboroda, Svitlana; Poggi, Bruno Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates In: Archive for Rational Mechanics and Analysis, vol. 248, no. 31, 2024. Abstract | Links @article{nokey,
title = {Critical Perturbations for Second Order Elliptic Operators. Part II: Non-tangential maximal function estimates},
author = {Simon Bortz and Steve Hofmann and José Luis Luna and Svitlana Mayboroda and Bruno Poggi},
url = {https://arxiv.org/abs/2302.02746},
doi = {10.1007/s00205-024-01977-x},
year = {2024},
date = {2024-04-04},
urldate = {2024-02-19},
journal = {Archive for Rational Mechanics and Analysis},
volume = {248},
number = {31},
abstract = {This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-operatorname{div} A nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi-Nash-Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-$L^p$ ``$N<S$'' estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full $L^2$ bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class.
As a corollary, we claim the first results in an unbounded domain concerning the $L^p$-solvability of boundary value problems for the magnetic Schr"odinger operator $-(nabla-i{bf a})^2+V$ when the magnetic potential ${bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space.},
howpublished = {To appear in Archive for Rational Mechanics and Analysis},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-operatorname{div} A nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the $L^2$ well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi-Nash-Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-$L^p$ ``$N<S$'' estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full $L^2$ bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class.
As a corollary, we claim the first results in an unbounded domain concerning the $L^p$-solvability of boundary value problems for the magnetic Schr"odinger operator $-(nabla-i{bf a})^2+V$ when the magnetic potential ${bf a}$ and the electric potential $V$ are accordingly small in the norm of a scale-invariant Lebesgue space. |
Poggi, Bruno Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields To appear in Advances in Mathematics, 2024. Abstract | Links @unpublished{nokey,
title = {Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields},
author = {Bruno Poggi},
url = {https://arxiv.org/abs/2107.14103},
year = {2024},
date = {2024-04-03},
abstract = {We resolve both a conjecture and a problem of Z. Shen from the 90's regarding non-asymptotic bounds on the eigenvalue counting function of the magnetic Schr"odinger operator $L_$, under a mild decay condition on the Green's function. For $L_V$, we prove a priori exponential decay of Green's function, eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight $1/u$, which may degenerate. Similar a priori results hold for $L_{{bf a},V}$. Furthermore, when $ngeq3$ and $V$ satisfies a scale-invariant Kato condition and a weak doubling property, we show that $1/sqrt u$ is pointwise equivalent to the Fefferman-Phong-Shen maximal function $m(cdot,V)$ (also known as Shen's critical radius function); in particular this gives a setting where the Agmon distance with weight $1/u$ is not too degenerate. Finally, we extend results from the literature for $L_{{bf a},V}$ regarding exponential decay of the fundamental solution and eigenfunctions, to the situation of irregular magnetic fields with directionality assumptions.},
howpublished = {To appear in Advances in Mathematics},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We resolve both a conjecture and a problem of Z. Shen from the 90's regarding non-asymptotic bounds on the eigenvalue counting function of the magnetic Schr"odinger operator $L_$, under a mild decay condition on the Green's function. For $L_V$, we prove a priori exponential decay of Green's function, eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight $1/u$, which may degenerate. Similar a priori results hold for $L_{{bf a},V}$. Furthermore, when $ngeq3$ and $V$ satisfies a scale-invariant Kato condition and a weak doubling property, we show that $1/sqrt u$ is pointwise equivalent to the Fefferman-Phong-Shen maximal function $m(cdot,V)$ (also known as Shen's critical radius function); in particular this gives a setting where the Agmon distance with weight $1/u$ is not too degenerate. Finally, we extend results from the literature for $L_{{bf a},V}$ regarding exponential decay of the fundamental solution and eigenfunctions, to the situation of irregular magnetic fields with directionality assumptions. |
Chamorro, Diego; Poggi, Bruno On an almost sharp Liouville type theorem for fractional Navier-Stokes equations To appear in Publicacions Matemàtiques, 2024. Abstract | Links @unpublished{nokey,
title = {On an almost sharp Liouville type theorem for fractional Navier-Stokes equations},
author = {Diego Chamorro and Bruno Poggi},
url = {https://arxiv.org/abs/2211.13077},
year = {2024},
date = {2024-04-03},
abstract = {We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Delta)^{frac{alpha}{2}}$ with $0<alpha<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $dot{H}^{frac{alpha}{2}}(R)$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $alpha$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<alpha<5/3$. Moreover, in the case $1<alpha<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<alphaleq 1$ seems for the moment to be an open problem.},
howpublished = {To appear in Publicacions Matemàtiques},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We investigate existence, Liouville type theorems and regularity results for the 3D stationary and incompressible fractional Navier-Stokes equations: in this setting the usual Laplacian is replaced by its fractional power $(-Delta)^{frac{alpha}{2}}$ with $0<alpha<2$. By applying a fixed point argument, weak solutions can be obtained in the Sobolev space $dot{H}^{frac{alpha}{2}}(R)$ and if we add an extra integrability condition, stated in terms of Lebesgue spaces, then we can prove for some values of $alpha$ that the zero function is the unique smooth solution. The additional integrability condition is almost sharp for $3/5<alpha<5/3$. Moreover, in the case $1<alpha<2$ a gain of regularity is established under some conditions, however the study of regularity in the regime $0<alphaleq 1$ seems for the moment to be an open problem. |
Vol. 2023
|
Fleschler, Ian; Tolsa, Xavier; Villa, Michele Carleson's ε^2 conjecture in higher dimensions 2023. Abstract | Links @unpublished{nokey,
title = { Carleson's ε^2 conjecture in higher dimensions },
author = {Ian Fleschler and Xavier Tolsa and Michele Villa },
url = {https://arxiv.org/abs/2310.12316},
year = {2023},
date = {2023-10-18},
urldate = {2023-10-18},
abstract = { We prove a higher dimensional analogue of Carleson's ε^2 conjecture.},
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
We prove a higher dimensional analogue of Carleson's ε^2 conjecture. |
Fleschler, Ian; Tolsa, Xavier; Villa, Michele Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's ε2 conjecture in higher dimensions 2023. Abstract | Links @unpublished{nokey,
title = {Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's ε2 conjecture in higher dimensions},
author = {Ian Fleschler and Xavier Tolsa and Michele Villa},
url = {https://arxiv.org/abs/2306.06187},
year = {2023},
date = {2023-06-09},
urldate = {2023-06-09},
abstract = {The main aim of this article is to prove quantitative spectral inequalities for the Laplacian with Dirichlet boundary conditions. More specifically, we prove sharp quantitative stability for the Faber-Krahn inequality in terms of Newtonian capacities and Hausdorff contents of positive codimension, thus providing an answer to a question posed by De Philippis and Brasco.
One of our results asserts that for any bounded domain Ω⊂ℝn, n≥3, with Lebesgue measure equal to that of the unit ball and whose first eigenvalue is λΩ, denoting by λB the first eigenvalue for the unit ball, for any a∈(0,1) it holds
λΩ−λB≥C(a)infB(supt∈(0,1)1Hn−1(∂((1−t)B))∫∂((1−t)B)Capn−2(B(x,atrB)∖Ω)(trB)n−3dHn−1(x))2,
where the infimum is taken over all balls B with the same Lebesgue measure as Ω and Capn−2 is the Newtonian capacity of homogeneity n−2. In fact, this holds for bounded subdomains of the sphere and the hyperbolic space, as well.
In a second result, we also apply the new Faber-Krahn type inequalities to quantify the Hayman-Friedland inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's ε2-conjecture to higher dimensions in terms of a square function involving the characteristics of certain spherical domains, and we prove the necessity of the finiteness of such square function in the tangent points via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answer in the negative a question posed by Allen, Kriventsov and Neumayer in connection to rectifiability and the positivity set of the ACF monotonicity formula. },
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
The main aim of this article is to prove quantitative spectral inequalities for the Laplacian with Dirichlet boundary conditions. More specifically, we prove sharp quantitative stability for the Faber-Krahn inequality in terms of Newtonian capacities and Hausdorff contents of positive codimension, thus providing an answer to a question posed by De Philippis and Brasco.
One of our results asserts that for any bounded domain Ω⊂ℝn, n≥3, with Lebesgue measure equal to that of the unit ball and whose first eigenvalue is λΩ, denoting by λB the first eigenvalue for the unit ball, for any a∈(0,1) it holds
λΩ−λB≥C(a)infB(supt∈(0,1)1Hn−1(∂((1−t)B))∫∂((1−t)B)Capn−2(B(x,atrB)∖Ω)(trB)n−3dHn−1(x))2,
where the infimum is taken over all balls B with the same Lebesgue measure as Ω and Capn−2 is the Newtonian capacity of homogeneity n−2. In fact, this holds for bounded subdomains of the sphere and the hyperbolic space, as well.
In a second result, we also apply the new Faber-Krahn type inequalities to quantify the Hayman-Friedland inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's ε2-conjecture to higher dimensions in terms of a square function involving the characteristics of certain spherical domains, and we prove the necessity of the finiteness of such square function in the tangent points via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answer in the negative a question posed by Allen, Kriventsov and Neumayer in connection to rectifiability and the positivity set of the ACF monotonicity formula. |
Gallegos, Josep M.; Mourgoglou, Mihalis; Tolsa, Xavier Extrapolation of solvability of the regularity problem in rough domains 2023. Abstract | Links @unpublished{nokey,
title = { Extrapolation of solvability of the regularity problem in rough domains},
author = {Josep M. Gallegos and Mihalis Mourgoglou and Xavier Tolsa },
url = {https://arxiv.org/abs/2310.12316},
year = {2023},
date = {2023-06-09},
urldate = {2023-06-09},
abstract = { Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with compact and uniformly n-rectifiable boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,p(∂Ω) for p>1 implies solvability in M1,1(∂Ω) and, in the particular case of the Laplacian, solvability in M1,1(∂Ω) implies solvability in M1,p(∂Ω) for some p>1. Moreover, under the hypothesis that ∂Ω supports a 1-weak Poincaré inequality, we prove that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives. },
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
Let Ω⊂Rn+1, n≥2, be an open set satisfying the corkscrew condition with compact and uniformly n-rectifiable boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M1,1(∂Ω) and the weak-A∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M1,p(∂Ω) for p>1 implies solvability in M1,1(∂Ω) and, in the particular case of the Laplacian, solvability in M1,1(∂Ω) implies solvability in M1,p(∂Ω) for some p>1. Moreover, under the hypothesis that ∂Ω supports a 1-weak Poincaré inequality, we prove that the solvability of the regularity problem in the Hajłasz-Sobolev space M1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives. |
Gallegos, Josep Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant In: Calculus of Variations and Partial Differential Equations, vol. 62, 2023. Abstract | Links @article{nokey,
title = {Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant},
author = {Josep Gallegos},
doi = {https://doi.org/10.1007/s00526-022-02426-x},
year = {2023},
date = {2023-03-17},
journal = {Calculus of Variations and Partial Differential Equations},
volume = {62},
abstract = {Let Ω⊂Rd be a C1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d×d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div(A(x)∇u)=0 in Ω, and u vanishes on Σ=∂Ω∩B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i such that u|Bi∩Ω does not change sign and K∖⋃iBi has Minkowski dimension smaller than d−1−ϵ for any compact K⊂Σ. We also find upper bounds for the (d−1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Σ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Σ is bounded except for a set of Hausdorff dimension at most d−1−ϵ. },
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Let Ω⊂Rd be a C1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d×d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div(A(x)∇u)=0 in Ω, and u vanishes on Σ=∂Ω∩B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i such that u|Bi∩Ω does not change sign and K∖⋃iBi has Minkowski dimension smaller than d−1−ϵ for any compact K⊂Σ. We also find upper bounds for the (d−1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Σ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Σ is bounded except for a set of Hausdorff dimension at most d−1−ϵ. |
Tolsa, Xavier The dimension of harmonic measure on some AD-regular flat sets of fractional dimension To appear in Int. Math. Res. Not. IMRN , 2023. Abstract | Links @unpublished{nokey,
title = {The dimension of harmonic measure on some AD-regular flat sets of fractional dimension},
author = {Xavier Tolsa},
url = {https://arxiv.org/abs/2301.04084},
year = {2023},
date = {2023-01-20},
urldate = {2023-01-20},
abstract = { In this paper it is shown that if E⊂Rn+1 is an s-AD regular compact set, with s∈[n−12,n), and E is contained in a hyperplane or, more generally, in an n-dimensional C1 manifold, then the Hausdorff dimension of the harmonic measure for the domain R^{n+1}∖E is strictly smaller than s, i.e., than the Hausdorff dimension of E. },
howpublished = {To appear in Int. Math. Res. Not. IMRN },
keywords = {},
pubstate = {published},
tppubtype = {unpublished}
}
In this paper it is shown that if E⊂Rn+1 is an s-AD regular compact set, with s∈[n−12,n), and E is contained in a hyperplane or, more generally, in an n-dimensional C1 manifold, then the Hausdorff dimension of the harmonic measure for the domain R^{n+1}∖E is strictly smaller than s, i.e., than the Hausdorff dimension of E. |
