# Questions tagged [existence-theorems]

The tag has no usage guidance.

53 questions
Filter by
Sorted by
Tagged with
1 vote
47 views

• 41
121 views

### Conditions replacing compactness

Reading this book, the authors used the following "classic" idea: Let $X$ be a Banach space and $C$ a nonempty, weakly compact, convex subset of $X$. Let $T: C \rightarrow C$ be a ...
• 291
40 views

### Existence and Uniqueness of lifting Hele-Shaw problem

I am researching for the existence and uniqueness of solutions for the equation in figure below enter image description here $$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$ The ...
656 views

### What is the most general Carathéodory-type global existence theorem?

I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$ \left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \in ...
• 182
1 vote
102 views

### Existence theory for geometric flow of space curves

Is there any existence theory applicable to general geometric flows of space curves in the following form? $$\partial_t \gamma = v_t t + v_n n + v_b b$$ Here $\gamma$ is the evolving curve, $t$, $n$ ...
• 11
53 views

### Global existence for one Cauchy problem based on global existence of other two auxiliary Cauchy problems

I have a Cauchy problem for the differential equation $$y' = f(t, y),$$ with initial condition $y(0) = y^0$; here, $y$ and $f$ are two-dimensional vector-functions. The ...
• 19
1 vote
84 views

• 101
77 views

### Existence of a shift invariant selection map

Some time ago I asked this question, but now realise that it is harder than I anticipated. Therefore I am taking a step back to the following problem. Let $X$ and $Y$ be two sets and $F$ a point to ...
• 479
1 vote
124 views

• 143
111 views

• 191
1k views

### Global Solutions of Ordinary Differential Equations

Background Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying, $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$, for every ...
• 123
101 views

### reference on existence result for nonlinear elliptic PDE

During my work, I came to the question of existence of weak solutions to the following elliptic equation $$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$ with ...
• 51
281 views

### Existence and uniqueness of heteroclinic orbits

I am looking for conditions on a nonlinear dynamical system $\frac{d\vec{x}}{dt} = \vec{F}(\vec{x})$ that guarantee the existence of a unique heteroclinic orbit between a stable attractor of this ...
• 229
What are modular forms or cusps forms, resp. ? We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The ...