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36

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Introduction

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January 2011 - present

January 2009 - December 2010

## Publications

Publications (36)

This work extends Perron's method for the porous medium equation in the slow
diffusion case. The main result shows that nonnegative continuous boundary
functions are resolutive in a general cylindrical domain.

We prove the uniqueness for viscosity solutions of a differential equation
involving the infinity-Laplacian with a variable exponent. A version of the
Harnack's inequality is derived for this minimax problem.

This article describes modal analysis of acoustic waves in the human vocal tract while the subject is pronouncing [o]. The model used is the wave equation in three dimensions, together with physically relevant boundary conditions. The geometry is reconstructed from anatomical MRI data obtained by other researchers. The computations are carried out...

We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous, provided that the obstacle is Hölder continuous.

We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.

We prove that various notions of supersolutions to the porous medium equation are equivalent under suitable conditions. More spesifically, we consider weak supersolutions, very weak supersolutions, and $m$-superporous functions defined via a comparison principle. The proofs are based on comparison principles and a Schwarz type alternating method, w...

We prove a comparison principle for the porous medium equation in more
general open sets in $\mathbb{R}^{n+1}$ than space-time cylinders. We apply
this result in two related contexts: we establish a connection between a
potential theoretic notion of the obstacle problem and a notion based on a
variational inequality. We also prove the basic propert...

We prove existence results for the obstacle problem related to the porous medium equation. For sufficiently regular obstacles, we find continuous solutions whose time derivative belongs to the dual of a parabolic Sobolev space. We also employ the notion of weak solutions and show that for more general obstacles, such a weak solution exists. The lat...

We consider a generalised Webster's equation for describing wave propagation
in curved tubular structures such as variable diameter acoustic wave guides.
Webster's equation in generalised form has been rigorously derived in a
previous article starting from the wave equation, and it approximates
cross-sectional averages of the propagating wave. Here...

We prove that solutions to Cauchy problems related to the $p$-parabolic
equations are stable with respect to the nonlinearity exponent $p$. More
specifically, solutions with a fixed initial trace converge in an $L^q$-space
to a solution of the limit problem as $p>2$ varies.

Weak supersolutions to the porous medium equation are defined by means of
smooth test functions under an integral sign. We show that nonnegative weak
supersolutions become lower semicontinuous after redefinition on a set of
measure zero. This shows that weak supersolutions belong to a class of
supersolutions defined by a comparison principle.

We prove optimal integrability results for solutions of the
$p(\cdot)$-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain
this, we show that variable exponent Riesz and Wolff potentials maps $L^1$ to
variable exponent weak Lebesgue spaces.

We study the regularity properties of solutions to the single and double obstacle problem with non standard growth. Our main results are a global reverse Hölder inequality, Hölder conti-nuity up to the boundary, and stability of solutions with respect to continuous perturbations in the variable growth exponent.

We prove the unique solvability, passivity/conservativity and some regularity
results of two mathematical models for acoustic wave propagation in curved,
variable diameter tubular structures of finite length. The first of the models
is the generalised Webster's model that includes dissipation and curvature of
the 1D waveguide. The second model is t...

We prove that arbitrary superharmonic functions and superparabolic functions
related to the $p$-Laplace and the $p$-parabolic equations are locally obtained
as limits of supersolutions with desired convergence properties of the
corresponding Riesz measures. As an application we show that a family of
uniformly bounded supersolutions to the $p$-parab...

We prove stability results for nonlinear diffusion equations of the porous
medium and fast diffusion types with respect to the nonlinearity power $m$:
solutions with fixed data converge in a suitable sense to the solution of the
limit problem with the same data as $m$ varies. Our arguments are elementary
and based on a general principle. We use nei...

We show the existence of solutions to the fast diffusion equation with a general finite and positive Borel measure as the
right hand side source term.
Mathematics Subject Classification (2010)Primary 35K67–Secondary 35R06–35A01–35B45

Wave propagation in curved tubular domains is considered. A general version
of Webster's equation is derived from the scattering passive wave equation.
More precisely, it is shown that planar averages of a sufficiently smooth
solution of the wave equation satisfy the corresponding Webster's equation when
the latter includes additional control signa...

We study the regularity properties of solutions to el-liptic equations similar to the p(·)-Laplacian. Our main results are a global reverse Hölder inequality, Hölder continuity up to the boundary, and stability of solutions with respect to continuous per-turbations in the variable growth exponent. We assume that the complement of the domain is unif...

We study superharmonic functions related to elliptic equations with structural conditions involving a variable growth exponent. We establish pointwise estimates for such functions in terms of a Wolff type potential. We apply these estimates to prove a variable exponent version of the Hedberg–Wolff theorem on the dual of Sobolev spaces with zero bou...

We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right-hand
side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower
semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutio...

We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency
of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity
involving the variable growth exponent. We also prove the Hölder continuity of weak solut...

We consider different notions of solutions to the p(lambda) Laplace equation -div(vertical bar Du(x)vertical bar(p(lambda)-2) Du(x)) = 0 with 1 < p(x) < infinity We show by proving a comparison principle that viscosity supersolutions and p(x)-superharmonic functions of nonlinear potential theory coincide This implies that weak and viscosity solutio...

We study superparabolic functions related to nonlinear parabolic equations. They are defined by means of a parabolic comparison principle with respect to solutions. We show that every superparabolic function satisfies the equation with a positive Radon measure on the right-hand side, and conversely, for every finite positive Radon measure there exi...

We prove a removability result for nonlinear elliptic equations withp (x)-type nonstandard growth and estimate the growth of solutions near a nonremovable isolated singularity. To accomplish this, we employ a Harnack estimate for possibly unbounded solutions and the fact that solutions with nonremovable isolated singularities are p (x)-superharmoni...

We study the balayage related to the supersolutions of the variable exponent p(·)-Laplace equation. We prove the fundamental convergence theorem for the balayage and apply it for proving the Kellogg property,
boundary regularity results for the balayage, and a removability theorem for p(·)-solutions.
KeywordsNon-standard growth–
p(·)-Laplacian–Com...

We prove Harnack inequalities for quasiminimizers of the variable exponent Dirichlet energy integral by employing the De Giorgi method.

We show that given a positive and finite Radon measure $\mu$, there is a $\Apx$ -superharmonic function $u$ which satisfies
$-\dive\A(x,Du)=\mu$
¶ in the sense of distributions. Here $\A$ is an elliptic operator with $p(x)$-type nonstandard growth.

We discuss recording arrangements for speech during an MRI scan of the speak-ers vocal tract. The image and sound data thus obtained will be used for construction and vali-dation of a numerical model for the vocal tract.

A. In this article we study solutions and supersolutions of a vari-able exponent p(·)-Laplace equation, and the corresponding obstacle prob-lem, as well as related p(·)-superharmonic functions. The relationship between these function classes closely parallels the classical case. How-ever, integrability properties of p(·)-superharmonic functi...

We show that every weak supersolution of a variable exponent p-Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically HÃƒÂ¶lder continuous. As a technical tool we derive Harnack-type estimates for possibly unbounded supersolutions.

We study computationally the dynamics of sound production in the vocal tract (VT). Our mathemat- ical formulation is based on the three-dimensional wave equation, together with physically relevant boundary conditions. We focus on formant and pressure information in the VT. For this purpose, we make use of anatomical data obtained by MRI by other re...

A parametric synthesis model for wind chimes is presented. A physics-based stochastic process is used to trigger a modal sound-production model for the tubes of a wind chime. Mode parameters are extracted in a semiautomatic fashion from a sound sample. An example and some possible applications are presented.

We consider the nonlinear potential theory of elliptic partial differential equations with nonstandard structural conditions. In such a theory, Harnack inequalities and the class of superharmonic functions related to the equation under consideration have a crucial role. We develop a technique for proving Harnack type inequalities to handle possibly...

## Projects

Project (1)