Mathematics

PUBLICATIONS

PDF files: [ Research statement ] [ List of publications ] [ List with abstracts ]

Abstracts: [ show all ] [ hide all ]

---

13. Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane. arXiv:1009.5326
    (with Richard Laugesen) Abstract

    Let $\lambda_i$ $(\mu_i)$ denote the eigenvalues of the Dirichlet (Neumann) Laplacian on a domain. Let $D$ be an $n$-fold rotationally symmetric domain and $T$ a linear transformation. Among the domains $T(D)$ the Laplace eigenvalues are maximal for the initial domain $D$. We use area $A$ and moment of inertia $I$ as scaling factors. Precisely $$\left.(\lambda_1+\cdots+\lambda_n){A^3\over I}\right|_{T(D)}$$ is maximal when $T$ is a multiple of identity.
    $\qquad$ We use the tight frame generated by the symmetry groups of $D$ to simplify Rayleigh quotients on $T(D)$. Our method does not need the exact eigenfunctions of $D$. We also treat Robin boundary conditions and certain Schrödinger operators.

12. Dirichlet eigenvalue sums on triangles are minimal for equilaterals. arXiv:1008.1316
    (with Richard Laugesen) Abstract

    Let $\lambda_i$ denote the eigenvalues of the Dirichlet Laplacian on an arbitrary triangle. Denote the diameter of this triangle by $D$. We show that among triangles the scale invariant quantity $$(\lambda_1+\cdots+\lambda_n)D^2\text{ is minimal for equilateral triangles,}$$ for any $n>1$. We also show that $$\lambda_1D^2,\;\lambda_2D^2,\; \lambda_3D^2\text{ are minimal for equilateral triangles.}$$ This suggests that all eigenvalues should be minimal for equilateral triangles.
    $\qquad$ We use unknown eigenfunctions of triangles as test functions on equilateral and right isosceles triangles (known cases). Later, we interpolate between the known cases.

11. Isoperimetric inequalities for eigenvalues of triangles. preprint and [extended version arXiv:0707.3631]
    Indiana Univ. Math. J., 59, 2010 Abstract

    We prove new isoperimetric bounds for eigenvalues of triangles using continuous Steiner symmetrization and polarization. These results can viewed as generalizations of Pólya and Szegö's isoperimetric bound for triangles. Consider an isosceles triangle $I$ with the same area and the same smallest angle as a given triangle $T$. Consider also a circular sector $S$ with the same area and the same smallest angle. We have $$\lambda_1(T)\ge\lambda_1(I)\ge\lambda_1(S).$$ Therefore one can symmetrize a triangle to an isosceles triangle and this new triangles can be further symmetrized into a circular sector (with explicitly known eigenvalue). We also show the the eigenvalues of isosceles triangles with fixed area are unimodal with respect to the aperture angle with minimum at the equilateral triangle.
    $\qquad$We also show that among triangles with fixed inradius $R$ \begin{align} \lambda_1R^2|_{Triangle}&\le\lambda_1R^2|_{Equilateral}\\ (\lambda_2-\lambda_1)R^2|_{Triangle}&\le(\lambda_2-\lambda_1)R^2|_{Equilateral}\\ \left.{\lambda_2\over\lambda_1}\right|_{Acute\;Triangle}&\le\left.{\lambda_2\over\lambda_1}\right|_{Equilateral} \end{align} The last result resembles the Payne-Pólya-Weinberger conjecture proved by Ashbaugh and Benguria.

10. Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. MR2644129
    (with Richard Laugesen) J. Differential Equations, bf 249 (2010), no. 1, 118-135. Abstract

    Let $\mu_i$ denote the eigenvalues of the Neumann Laplacian on a triangular domain and let $D$ be the diameter of the domain. We show that $$\mu_1D^2\text{ is minimal for degenerate acute isosceles triangles.}$$ As a corollary we get the optimal Poincaré inequality for triangles: $$\int_T v^2\,dA<\frac{D^2}{j_{1,1}^2}\int_T |\nabla v|^2\, dA$$ To prove the result we first show that the fundamental mode is symmetric on isosceles triangles with aperture angle less than $\pi/3$ and antisymmetric otherwise.
    $\qquad$ We use the Method of the Unknown Trial Function on isosceles triangles by "interpolating" between equilateral triangles and circular sectors.

9. Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. MR2674631
   (with Pedro Freitas) ESAIM Control Optim. Calc. Var. 16(3):648-676, 2010 Abstract

   We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. We divide them into simple (some of them presented below) and functional (more complicated). We also give extensible comparisons for the known bounds.

   • Quadrilaterals: We improve Pólya and Szegö's 1951 isoperimetric lower bound for quadrilaterals. We also extend Hersch's 1966 upper bound for parallelograms to general quadrilaterals. $${2\pi^2\over A}+{\pi^2\over 4A^2}(d_2\sin\alpha-d_1)^2\le\lambda_1(Q)\le{\pi^2\over2A^2}(l_1^2+l_2^2+l_3^2+l_4^2)$$ Here $A$ denotes the area, $d_1$ and $d_2$ the lengths of the diagonals and $\alpha$ the angle between them. Finally $l_i$ are the lengths of the sides of quadrilateral $Q$. The equality holds for squares in the lower bound and for all rectangles in the upper bound.
     $\qquad$ We also show that for a rhombus with diagonals $d$ and $ad$ with $0<a<1$ $${2\pi^2(1+a^2)\over a^2d^2}-{\pi^2(1-a)^2\over a^2d^2}\le\lambda_1(R)\le{2\pi^2(1+a^2)\over a^2d^2}-0.796{\pi^2(1-a)^2\over a^2d^2}$$ The lower bound is due to Hooker and Protter.
   • Triangles: Let $d$ and $h$ denote the diameter and the shortest altitude of a triangle. Put $\beta=\arcsin(d/h)$. Then $$\pi^2\left(\frac1d+\frac1h\right)^2\le\lambda_1(T)\le {j_{\pi/\beta}^2\over A}\tan(\beta/2)$$
8. Eigenvalue inequalities for mixed Steklov problems. arXiv:0909.5473 [AMS bookstore]
   (with Rodrigo Bañuelos, Tadeusz Kulczycki, and Iosif Polterovich)
   Operator Theory and Its Applications, In Memory of V. B. Lidskii (1924-2008), Amer. Math. Soc. Transl., 231, 2010 Abstract

   Consider a well $W$ in $R^{d}$ with a flat opening $F\subset R^{d-1}$. Assume that this well satisfies "weak" John's condition. Put Steklov boundary conditions on the opening and either Dirichlet or Neumann on the sides $B$ of the well. We prove an inequality between mixed Steklov-Dirichlet eigenvalues $\lambda_i$ and mixed Steklov-Neumann eigenvalues $\mu_i$ $$\mu_{i+1}\le\lambda_i,\text{ for any }i\ge1.$$ The inequality is strict if $d > 3$. A stronger inequality holds if the well satisfies the standard John's condition. This result generalizes classical inequalities between Dirichlet and Neumann eigenvalues.
   $\qquad$ We also show that the nodal set of the eigenfunction belonging to $\mu_2$ must intersect $B$.
   

   An example of a well that satisfies "weak" John condition, but not classical John condition. It is "almost" contained in a straight infinite well $F\times(-\infty,0]$.

   
   
7. Maximizing Neumann fundamental tones of triangles. MR2567204
   (with Richard Laugesen) J. Math. Phys., 50:112903, 2009 Abstract

   Consider the eigenvalues $\mu_i$ of the Neumann Laplacian on triangular domains. We prove bounds for the first two nonzero eigenvalues involving various geometric measurements of the domain (area $A$, perimeter $L$, moment of inertia $I$). In particular $$ \mu_1\frac{I}A\le\frac{16\pi^2}{3},\qquad\qquad\qquad \mu_1L^2\le16\pi^2,\qquad\qquad\qquad \mu_1A\le\frac{4\pi^2}{3\sqrt{3}}. $$ The first bound is the strongest and in all of them the equality holds only for equilateral triangles. All these bounds can be derived from the following bound involving isoperimetric excess $E=\frac{L^2}{12\sqrt{3}}-A$ (see also [3]): $$\mu_1\left(A+\frac{\pi^2}{j_{0,1}^2}E\right)\le \frac{4\pi^2}{3\sqrt{3}}.$$ Note that equality holds for equilateral triangles and asymptotically for degenerate obtuse isosceles triangles.
   $\qquad$ We also show that scaled averages (harmonic $H$ and arithmetic $M$) of the first two eigenvalues $$H(\mu_1,\mu_2)A\text{ and }M(\mu_1,\mu_2)\frac{A^3}{I}\text{ are both maximal for equilateral triangles.}$$ Note that the arithmetic average result is a special case of our later general results from [13].

6. On the trace of symmetric stable processes on Lipschitz domains. MR2465712
   (with Rodrigo Bañuelos and Tadeusz Kulczycki) Potential Anal., 30(1):65-83, 2009. Abstract

   Let $X_t$ be a symmetric $\alpha$-stable process in $R^d$, $\alpha \in (0,2]$. This is a process with independent and stationary increments and characteristic function $E^0 e^{i \xi X_t} = e^{-t |\xi|^{\alpha}}$, $\xi \in R^d$, $ t > 0$. By $p_D(t,x,y)$ we denote the transition density of this process starting at the point $x$ and killed outside of $D$.
   $\qquad$ The trace of the $\alpha$-stable heat kernel on $D$ (often referred to as the partition function of $D$) is defined by $$ Z_D(t)=\int_D p_D(t,x,x)dx.$$ We show that for Lipschitz domain $D$ $$ t^{d/\alpha}Z_D(t)= C_1|D|-C_2 {H}^{d-1}(\partial{D}) t^{1/\alpha}+o(t^{1/\alpha}),$$ where $|D|$ is the Lebesgue measure of the domain and ${H}^{d-1}(\partial{D})$ is the Hausdorff measure of its boundary.
   $\qquad$ Our result generalizes earlier results about smooth domains. Note also that similar formula is known for Brownian motion ($\alpha=2$). We use geometric properties of the boundary of a Lipschitz domain to reduce the problem to a half-space case. This reduction requires precise estimates on the heat kernel from my earlier paper [2].

5. Neumann Bessel heat kernel monotonicity. MR2568694
   (with Rodrigo Bañuelos and Tadeusz Kulczycki) J. Funct. Anal., 257(10):3329-3352, 2009. Abstract

   Consider a $d$-dimensional reflected Brownian motion $W_t^{B}$ on a unit ball $B$. The radial part of this process is a Bessel process $R_t$ on interval $(0,1]$ reflected at $1$. Let $p_I^R(t,r,\rho)$ be the transition probabilities (heat kernel) of $R_t$.
   $\qquad$ We show that the function $p_I^R(t,r,r)$ is increasing toward $1$ in dimensions $d > 2$, but not $d=2$. The result is motivated by the conjecture of Laugesen and Morpurgo stating that the Neumann heat kernel $p_B^N(t,x,x)$ is increasing toward the boundary of $B$.
   $\qquad$ We use discrete approximation for a reflected Brownian motion on the ball $B$. The random walk we use is tailored for the reflection on the surface of a ball. We allow unit transitions from $x$ along a radial direction $($to $U(x)$ and $D(x))$, and uniform jumps to a disk $C(x)$. A cross-section of a ball with transitions is showed on the figure.
   

   Transitions from point $x$ for a random walk well suited for radial reflections. Note that the length of this random walk is a random walk with unit forward and backward transitions.

   
   
4. Scattering length for stable processes. MR2524659
   Illinois J. Math., 52(2):667-680, 2008. Abstract

   Let $X_t$ be the isotropic $\alpha$-stable Lévy process in $R^d$ with the characteristic function $E^0(\exp(i\xi X_t))=\exp(-t|\xi|^\alpha)$. For simplicity we assume that $d > \alpha$. The generator of this process is the fractional Laplacian $\Delta^{\alpha/2}$. Consider also the reflected stable process $Y_t$ on $\Omega$ with the generator $\Delta^{\alpha/2}_N$ ("Neumann" fractional Laplacian).
   $\qquad$ For a positive function $v$ we define the capacitory potential, capacitory measure and scattering length \begin{align} U_v(x)&=1-E^x\exp\left(-\int_0^\infty v(X_s)ds\right),\\ \mu_v&=-\Delta^{\alpha/2} U_v,\\ \Gamma(v)&=\int_{R^d} d\mu_v(x). \end{align} We show that the smallest eigenvalue of the Schrödinger operator $\Delta^{\alpha/2}_N+v$ on a cube $\Omega$ satisfies $$C_1(\Omega)\Gamma(v)\leq \lambda_1(v)\leq C_2(\Omega)\Gamma(v).$$ The upper bound holds only if $\Gamma(v)$ is small enough.

3. Sharp bounds for eigenvalues of triangles. MR2369934
   Michigan Math. J., 55(2):243-254, 2007. Abstract

   Let $T$ be a triangle with area $A$ and perimeter $L$. Then the first eigenvalue $\lambda_T$ of the Dirichlet Laplacian on $T$ satisfies $$ {\pi^2 L^2\over 16A^2}<\lambda_T\leq {\pi^2 L^2\over 9A^2}.$$ The constants $9$ and $16$ are optimal, and equality in the upper bound holds only for the equilateral triangle. The lower bound was proved in a more general context by Makai. We show that for "tall" isosceles triangles there is an asymptotic equality in the lower bound. Hence it is impossible to decrease the constant $16$.
   $\qquad$ We combine known eigenfunctions of equilateral and some right triangles to get test functions for arbitrary triangles. Nearly degenerate cases are handled using comparisons with circular sectors.
   $\qquad$ We conjecture the following stronger bounds involving isoperimetric excess $E=\frac{L^2}{12\sqrt{3}}-A$ (see also [7]) $${4\pi^2\over \sqrt3A}+{3\sqrt{3}\pi^2E\over4A^2} ={\pi^2L^2\over16A^2}+{7\sqrt3\pi^2\over12A}\leq\lambda_T\leq {\pi^2L^2\over12A^2}+{\sqrt3\pi^2\over3A}= {4\pi^2\over \sqrt3A}+{\sqrt{3}\pi^2E \over A^2} .$$ Note that ${\pi^2 L^2\over 9A^2}={4\pi^2\over \sqrt3A}+{4\pi^2E \over \sqrt{3}A^2}$.

2. Symmetric stable processes on unbounded domains. MR2255353
   Potential Anal., 25(4):371-386, 2006. Abstract

   Let $X_t$ be the isotropic $\alpha$-stable Lévy process in $R^d$ with the characteristic function $E^0(\exp(i\xi X_t))=\exp(-t|\xi|^\alpha)$. Consider a nondecreasing function $f(t)$, and domain $D_f$ obtained by revolving $f$. Finally, let $M(t)$ be differentiable and nondecreasing with at most polynomial growth.
   $\qquad$ We give a condition for the $M$-moment of the exit time $\tau_f$ of $X_t$ from $D_f$ to be finite. $$ E^x(M(\tau_f))<\infty \;\; \iff \;\; \int_1^\infty {f^{d-1}(t)\over x^{d+\alpha}}\int_0^\infty M'(s) \exp\left(-s\over f^\alpha(t) \right)\;ds\,dt<\infty, $$ In particular $$ E^x\left(\tau_f^\beta\ln^\gamma(\tau_f)\right)<\infty \;\; \iff \;\; \int_1^\infty {f^{d+\alpha\beta-1}\!(t)\ln^{\gamma}f(t)\over t^{d+\alpha}}\;dt<\infty, $$ To prove the result we find a sharp upper bound for the heat kernel on convex domains. We say that a domain is semibounded with bound $R$ if it can be squeezed between two hyperplanes placed $2R$ apart. For such domains, there exist $\lambda,c>0$ such that $$ p_D(t,x,y)\leq c\exp\left(-\lambda t\over R^\alpha\right) p(t,x,y) \min\left\{1,{\delta_D^{\alpha/2}(y)\over \sqrt{t}}, {\delta_D^{\alpha/2}(x)\over \sqrt{t}}, {\delta_D^{\alpha/2}(y)\delta_D^{\alpha/2}(x)\over t}\right\}, $$ where $p(t,x,y)$ is the free kernel. Note that the formula holds even if $R=\infty$.

1. Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes. MR2231884
   (with Tadeusz Kulczycki) Trans. Amer. Math. Soc., 358(11):5025-5057 (electronic), 2006. Abstract

   Let $X_t$ be the relativistic $\alpha$-stable process in $R^d,\ \alpha\in(0,2),\ d>\alpha$, with infinitesimal generator $$H_0^{(\alpha)}=-((-\Delta+m^{2/\alpha})^{\alpha/2}-m).$$ Consider a nonnegative, locally bounded potential $V$ and a Feynman-Kac semigroup $T_t$ with generator $H_0^{(\alpha)}-V$. We show that if $\lim_{x\to\infty}V(x)=\infty$, then for every $t>0$ the operator $T_t$ is compact.
   $\qquad$ We also consider a family of nonnegative potentials $V$, such that $\lim_{x\to\infty}V(x)=\infty$ and $V$ is comparable with a radial and nondecreasing function. For such potentials the semigroup $T_t$ is intrinsically ultracontractive (IU) if and only if $\lim_{x\to\infty}V(x)/|x|=\infty$. If this condition is satisfied we also obtain a sharp estimate on the first eigenfunction $\varphi_1$ of $T_t$. In particular, for power potentials $V(x)=|x|^\beta$ with $\beta>1$ we get $$ \varphi_1(x)\asymp \exp\left(-m^{1/\alpha}|x|\right)\left(1+|x|\right)^{(-d-\alpha-2\beta-1)/2}. $$