Application of cold causes temporary numbness of the nerve endings, leading to reduced sensation, soothing the skin, and reducing your urge to itch. It also helps in sloughing off dead skin, leading to a smoother appearance and quicker healing.  Put cold water in a bowl with some bath oil. If you want it colder, you can add some ice to the water. Soak a paper towel or clean washcloth in the water. Hold it on your face over the affected area for about 5 minutes. A cream or ointment is best, as they have more oil than lotions, which are heavier on water. The oil helps protect and moisturize your skin better.  Pick a cream that is fragrance-free, and make sure to keep it out of your eyes when applying it. Apply moisturizers as often as your skin feels dry. Applying it after a shower or washing your face is especially effective. These moisturizers soften skin and help heal and prevent flare-ups. Stress can make your eczema worse, as can exposure to chemical irritants. Because of this, it often helps to use a holistic medicine approach. Aromatherapy, massage, and similar techniques can help reduce stress and potentially bolster your immune system. Many alternative medicine skin preparations are soothing and irritant-free, such as a light layer of unrefined coconut oil.  If you are currently taking medication for your eczema, consult your doctor before taking any dietary supplements or skin care treatments, including herbal remedies. Essential oils are highly concentrated and should not be used undiluted, especially around sensitive areas such as the eyes. Even when diluted, take care not to let any get in your eyes. Oral antibiotics are sometimes used when you get an infection related to your dermatitis. Because the eye area is more sensitive, your doctor may want to prescribe you an antibiotic if you develop dermatitis around one or both of your eyes.

Summary:
Use an ice pack or cold compress around your eyes. Apply a moisturizer to your face. Stay healthy and comfortable. Ask about oral antibiotics.