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Panu Lahti

Publications and source records attributed to Panu Lahti.

At least 55 records · Page 3Linked to original sources
arXiv2018

Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry

We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincaré inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincaré inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collection of measures $\mathcal{H}^{Q-1}\lfloor_{ΣE}$ corresponding to any collection of sets $E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $ΣE$, if the image surface $Σf(E)$ does not see the singular set of $f$ as a large set, then $f(E)$ is also of finite perimeter. Even in the standard Euclidean setting our results are more general than that of Kelly, and hence are new even in there.

math.MG↗
arXiv2018

Approximation of BV by SBV functions in metric spaces

In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we show that functions of bounded variation (BV functions) can be approximated in the strict sense and pointwise uniformly by special functions of bounded variation, without adding significant jumps. As a main tool, we study the variational 1-capacity and its BV analog.

math.MG↗
arXiv2018

Federer's characterization of sets of finite perimeter in metric spaces

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete metric spaces that are equipped with a doubling measure and support a Poincaré inequality, the "only if" direction was shown by Ambrosio (2002). By applying fine potential theory in the case $p=1$, we prove that the "if" direction holds as well.

math.MG↗
arXiv2018

Superminimizers and a weak Cartan property for $p=1$ in metric spaces

We study functions of least gradient as well as related superminimizers and solutions of obstacle problems in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show a standard weak Harnack inequality and use it to prove semicontinuity properties of such functions. We also study some properties of the fine topology in the case $p=1$. Then we combine these theories to prove a weak Cartan property of superminimizers in the case $p=1$, as well as a strong version at points of nonzero capacity. Finally we employ the weak Cartan property to show that any topology that makes the upper representative $u^{\vee}$ of every $1$-superminimizer $u$ upper semicontinuous in open sets is stronger (in some cases, strictly) than the $1$-fine topology.

math.MG↗
arXiv2018

A new Cartan-type property and strict quasicoverings when $p=1$ in metric spaces

In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we prove a new Cartan-type property for the fine topology in the case $p=1$. Then we use this property to prove the existence of $1$-finely open \emph{strict subsets} and \emph{strict quasicoverings} of $1$-finely open sets. As an application, we study fine Newton-Sobolev spaces in the case $p=1$, that is, Newton-Sobolev spaces defined on $1$-finely open sets.

math.MG↗
arXiv2017

The variational 1-capacity and BV functions with zero boundary values on metric spaces

In the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of BV functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV functions in the BV norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and BV analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

math.MG↗
arXiv2017

An Analog of the Neumann Problem for the $1$-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is $1$ in a neighborhood of a point on the boundary of the domain, then the solution is $-1$ in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.

math.MG↗
arXiv2017

Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \emph{boundary trace} of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of \emph{continuous} solutions can fail, even in the weighted Euclidean setting with Lipschitz weights.

math.AP↗
arXiv2017

Quasiopen sets, bounded variation and lower semicontinuity in metric spaces

In the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to $L^1$-convergence in every $1$-quasiopen set. To achieve this, we first prove a new characterization of the total variation in $1$-quasiopen sets. Then we utilize the lower semicontinuity to show that the variation measures of a sequence of functions of bounded variation converging in the strict sense are uniformly absolutely continuous with respect to the $1$-capacity.

math.MG↗
arXiv2016

Notions of Dirichlet problem for functions of least gradient in metric measure spaces

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and used in this paper include the inner perimeter measure of a domain.

math.AP↗
arXiv2016

A Federer-style characterization of sets of finite perimeter on metric spaces

In the setting of a metric space equipped with a doubling measure that supports a Poincaré inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the codimension one Hausdorff measure of the \emph{$1$-fine boundary} of the set's measure theoretic interior $I_E$ is finite.

math.MG↗
arXiv2016

Strict and pointwise convergence of BV functions in metric spaces

In the setting of a metric space $X$ equipped with a doubling measure that supports a Poincaré inequality, we show that if $u_i\to u$ strictly in $BV(X)$, i.e. if $u_i\to u$ in $L^1(X)$ and $\Vert Du_i\Vert(X)\to\Vert Du\Vert(X)$, then for a subsequence (not relabeled) we have $\widetilde{u}_i(x)\to \widetilde{u}(x)$ for $\mathcal H$-almost every $x\in X\setminus S_u$.

math.MG↗
arXiv2016

Strong approximation of sets of finite perimeter in metric spaces

In the setting of a metric space equipped with a doubling measure that supports a Poincaré inequality, we show that any set of finite perimeter can be approximated in the BV norm by a set whose topological and measure theoretic boundaries almost coincide. This result appears to be new even in the Euclidean setting. The work relies on a quasicontinuity-type result for BV functions proved by Lahti and Shanmugalingam (2016).

math.MG↗
arXiv2015

Lower semicontinuity for an integral functional in BV

We prove a lower semicontinuity result for a functional of linear growth initially defined by \[ \int_ΩF\left(\frac{dDu}{dμ}\right)\,dμ\] for $u\in BV(Ω;\mathbb{R}^N)$ with $Du\ll μ$. The positive Radon measure $μ$ is only assumed to satisfy $\mathcal L^n\ll μ$.

math.FA↗
arXiv2015

A notion of fine continuity for BV functions on metric spaces

In the setting of a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.

math.AP↗
arXiv2015

Fine properties and a notion of quasicontinuity for BV functions on metric spaces

On a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that given a BV function, discarding a set of small $1$-capacity makes the function continuous outside its jump set and ``one-sidedly" continuous in its jump set. We show that such a property implies, in particular, that the measure theoretic boundary of a set of finite perimeter separates the measure theoretic interior of the set from its measure theoretic exterior, both in the sense of the subspace topology outside sets of small $1$-capacity, and in the sense of $1$-almost every curve.

math.AP↗
arXiv2015

Trace theorems for functions of bounded variation in metric spaces

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality, and obtain $L^1$ estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'ya-type inequality for functions of bounded variation that vanish on a set of positive capacity.

math.MG↗