<p>Motivated by lower bound estimates of Payne and Escobar for the classical Steklov problem, we study sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces with smooth boundary. We prove that, if the Gaussian curvature satisfies <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mi>K</mi><mo>≥</mo><mo>−</mo><mi>α</mi></math></EquationSource><EquationSource Format="TEX">$K \ge -\alpha $</EquationSource></InlineEquation> and the geodesic curvature of the boundary satisfies <InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><msub><mi>k</mi><mi>g</mi></msub><mo>≥</mo><mi>c</mi><mo>></mo><mn>0</mn></math></EquationSource><EquationSource Format="TEX">$k_{g} \ge c > 0$</EquationSource></InlineEquation>, then the first eigenvalue of the Steklov-type eigenvalue problem satisfies an optimal inequality and equality holds only for the Euclidean disk. In particular, we obtain a sharp lower bound for the first eigenvalue of a Schrödinger-Steklov eigenvalue problem. If the Gaussian curvature satisfies <InlineEquation ID="IEq3"><EquationSource Format="MATHML"><math><mi>K</mi><mo>≥</mo><mo>−</mo><mi>α</mi></math></EquationSource><EquationSource Format="TEX">$K \ge -\alpha $</EquationSource></InlineEquation> and the geodesic curvature satisfies <InlineEquation ID="IEq4"><EquationSource Format="MATHML"><math><msub><mi>k</mi><mi>g</mi></msub><mo>≥</mo><mi>c</mi><mo>></mo><mn>0</mn></math></EquationSource><EquationSource Format="TEX">$k_{g} \ge c > 0$</EquationSource></InlineEquation>, then the first eigenvalue <InlineEquation ID="IEq5"><EquationSource Format="MATHML"><math><msub><mi>σ</mi><mn>1</mn></msub></math></EquationSource><EquationSource Format="TEX">$\sigma _{1}$</EquationSource></InlineEquation> satisfies <Equation ID="Equa"><EquationSource Format="MATHML"><math><msub><mi>σ</mi><mn>1</mn></msub><mo>+</mo><mfrac><mi>α</mi><msub><mi>σ</mi><mn>1</mn></msub></mfrac><mo>≥</mo><mi>c</mi><mo>,</mo></math></EquationSource><EquationSource Format="TEX">$ \sigma _{1} + \frac{\alpha}{\sigma _{1}} \ge c, $</EquationSource></Equation> where equality holds if and only if the surface is a Euclidean disk of radius <InlineEquation ID="IEq6"><EquationSource Format="MATHML"><math><mn>1</mn><mo stretchy="false">/</mo><mi>c</mi></math></EquationSource><EquationSource Format="TEX">$1/c$</EquationSource></InlineEquation> and <InlineEquation ID="IEq7"><EquationSource Format="MATHML"><math><mi>α</mi><mo>=</mo><mn>0</mn></math></EquationSource><EquationSource Format="TEX">$\alpha = 0$</EquationSource></InlineEquation>. We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2<i>c</i> under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.</p>

Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

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2026年7月22日

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Gunhee Cho,
Keomkyo Seo

摘要

Motivated by lower bound estimates of Payne and Escobar for the classical Steklov problem, we study sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces with smooth boundary. We prove that, if the Gaussian curvature satisfies $K \geq - α$ $K \ge -\alpha $ and the geodesic curvature of the boundary satisfies $k_{g} \geq c > 0$ $k_{g} \ge c > 0$, then the first eigenvalue of the Steklov-type eigenvalue problem satisfies an optimal inequality and equality holds only for the Euclidean disk. In particular, we obtain a sharp lower bound for the first eigenvalue of a Schrödinger-Steklov eigenvalue problem. If the Gaussian curvature satisfies $K \geq - α$ $K \ge -\alpha $ and the geodesic curvature satisfies $k_{g} \geq c > 0$ $k_{g} \ge c > 0$, then the first eigenvalue $σ_{1}$ $\sigma _{1}$ satisfies $σ_{1} + \frac{α}{σ_{1}} \geq c,$ $ \sigma _{1} + \frac{\alpha}{\sigma _{1}} \ge c, $ where equality holds if and only if the surface is a Euclidean disk of radius $1 / c$ $1/c$ and $α = 0$ $\alpha = 0$. We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2c under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.