<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be a finite Borel measure on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(-\pi ,\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Consider the one-dimensional Poisson equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$-u''=\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msup> <mi>u</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>=</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>, where equality holds in the sense of distributions, with Dirichlet boundary conditions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u(\pm \pi )=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mo>±</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we define measures that are transformations of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, we compare the convex integral means of the original solutions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u_\mu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>μ</mi> </msub> </math></EquationSource> </InlineEquation> and the transformed ones, and we prove the uniqueness of a solution that maximizes the convex integral means.</p>

Symmetrization of measures and the one-dimensional Poisson equation with Dirichlet boundary conditions

期刊论文
发表日期：

2026年7月31日

查看全文

Christos Papadimitriou

摘要

Let $\mu $ $μ$ be a finite Borel measure on $(-\pi ,\pi )$ $(- π, π)$ . Consider the one-dimensional Poisson equation $-u''=\mu $ $- u^{''} = μ$ , where equality holds in the sense of distributions, with Dirichlet boundary conditions $u(\pm \pi )=0$ $u (\pm π) = 0$ . In this paper, we define measures that are transformations of $\mu $ $μ$ , we compare the convex integral means of the original solutions $u_\mu $ $u_{μ}$ and the transformed ones, and we prove the uniqueness of a solution that maximizes the convex integral means.