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Questions tagged [uniqueness-theorems]

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Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by $$ f(x,y):= (x^+)^2 + (y^+)^2 $$ where $a^+ = \max\{a,0\}$ for any real number $a$. Given a Lipschitz regular domain $\Omega \...
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1answer
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Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations: $$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$ has a unique Caratheodory solution ...
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Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?

This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold: https://arxiv.org/abs/1709.10033 What's the current ...
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Existence and uniqueness of Nonogram Solutions

Some of you may have stumbled upon Nonograms, a popular puzzle. It starts with an empty $n\times n$ grid with squares that must be filled and some numbers for each column and row. If we read "$3,3,2$"...
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Uniqueness problem for an elliptic system

I want to prove the uniqueness of the solution of the following problem: $$\eqalign{ & - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr & u > 0 \text{ in } \Omega \cr & \...
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Existence and uniqueness of geodesics in low regularity

Consider a Riemannian manifold $(M,g)$. How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...