In this paper we give a characterization of Ulam stability for the linear partial differential operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D:C^{1}(\mathbb {R}^{2},X)\rightarrow C(\mathbb {R}^{2},X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>:</mo> <msup> <mi>C</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Du=au_{x}+bu_{y}+cu$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mi>u</mi> <mo>=</mo> <mi>a</mi> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>u</mi> <mi>y</mi> </msub> <mo>+</mo> <mi>c</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a,b,c\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and X is a Banach space over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation>. Moreover, we obtain the best Ulam constant of the operator D.

Best Ulam constant of a partial differential operator

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2026年4月13日

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Adela Novac,
Diana Otrocol,
Dorian Popa

摘要

In this paper we give a characterization of Ulam stability for the linear partial differential operator $D:C^{1}(\mathbb {R}^{2},X)\rightarrow C(\mathbb {R}^{2},X)$ $D : C^{1} (R^{2}, X) \to C (R^{2}, X)$ given by $Du=au_{x}+bu_{y}+cu$ $D u = a u_{x} + b u_{y} + c u$ , where $a,b,c\in \mathbb {R}$ $a, b, c \in R$ and X is a Banach space over $\mathbb {R}$ $R$ . Moreover, we obtain the best Ulam constant of the operator D.